# Data model#

The tskit library deals with sets of sampled genome sequences through storage and analysis of their shared genetic ancestry. This genealogical ancestry (sometimes known as an Ancestral Recombination Graph) is stored concisely in tskit in the “succinct tree sequence” format, which comprises a collection of easy-to-understand tables. This page documents the structure of the tables and encoding of table data, as well as the encoding of the correlated genetic trees that can be extracted from a tskit tree sequence.

We begin by defining the the structure of the tables in the Table definitions section. The Data encoding section then describe how data is stored in those tables (also see the File formats chapter). The Tree structure section then describes the encoding of the trees that are generated from the NodeTable and EdgeTable. Finally, we describe how genotype data arises from tree structure, especially how we can incorporate the idea of missing data.

## Table definitions#

### Table types#

A tree sequence can be stored in a collection of eight tables: Node, Edge, Individual, Site, Mutation, Migration, Population, and Provenance. The Node and Edge tables store the genealogical relationships that define the trees, and the Individual table describes how multiple genomes are grouped within individuals; the Site and Mutation tables describe where mutations fall on the trees; the Migration table describes how lineages move across space; and the Provenance table contains information on where the data came from. Only Node and Edge tables are necessary to encode the genealogical trees; Sites and Mutations are optional but necessary to encode polymorphism (sequence) data; the remainder are optional. In the following sections we define these components of a tree sequence in more detail.

#### Node Table#

A node defines a monoploid set of chromosomes (a “genome”) of a specific individual that was born at some time in the past: the set of chromosomes inherited from a particular one of the individual’s parents. (See Nodes, Genomes, or Individuals? for more discussion.) Every vertex in the marginal trees of a tree sequence corresponds to exactly one node, and a node may be present in many trees. The node table contains five columns, of which flags and time are mandatory:

Column

Type

Description

flags

uint32

Bitwise flags.

time

double

Birth time of node.

population

int32

Birth population of node.

individual

int32

The individual the node belongs to.

metadata

binary

Node Metadata.

The time column records the birth time of the individual in question, and is a floating point value. Similarly, the population column records the ID of the population where this individual was born. If not provided, population defaults to the null ID (-1). Otherwise, the population ID must refer to a row in the Population Table. The individual column records the ID of the Individual individual that this node belongs to. If specified, the ID must refer to a valid individual. If not provided, individual defaults to the null ID (-1).

The flags column stores information about a particular node, and is composed of 32 bitwise boolean values. Currently, the only flag defined is NODE_IS_SAMPLE = 1, which defines the sample status of nodes. Marking a particular node as a “sample” means, for example, that the mutational state of the node will be included in the genotypes produced by TreeSequence.variants().

Bits 0-15 (inclusive) of the flags column are reserved for internal use by tskit and should not be used by applications for anything other than the purposes documented here. Bits 16-31 (inclusive) are free for applications to use for any purpose and will not be altered or interpreteted by tskit.

See the Node requirements section for details on the properties required for a valid set of nodes.

For convenience, the text format for nodes decomposes the flags value into its separate values. Thus, in the text format we have a column for is_sample, which corresponds to the flags column in the underlying table. As more flags values are defined, these will be added to the text file format.

The metadata column provides a location for client code to store information about each node. See the Metadata section for more details on how metadata columns should be used.

Note

The distinction between flags and metadata is that flags holds information about a node that the library understands, whereas metadata holds information about a node that the library does not understand. Metadata is for storing auxiliarly information that is not necessary for the core tree sequence algorithms.

#### Individual Table#

An individual defines how nodes (which can be seen as representing single chromosomes) group together in a polyploid individual. The individual table contains three columns, of which only flags is mandatory.

Column

Type

Description

flags

uint32

Bitwise flags.

location

double

Location in arbitrary dimensions.

parents

int32

Ids of parent individuals.

metadata

binary

Individual Metadata.

See the Individual requirements section for details on the properties required for a valid set of individuals.

The flags column stores information about a particular individual, and is composed of 32 bitwise boolean values. Currently, no flags are defined.

Bits 0-15 (inclusive) of the flags column are reserved for internal use by tskit and should not be used by applications for anything other than the purposes documented here. Bits 16-31 (inclusive) are free for applications to use for any purpose and will not be altered or interpreteted by tskit.

The location column stores the location of an individual in arbitrary dimensions. This column is ragged, and so different individuals can have locations with different dimensions (i.e., one individual may have location [] and another [0, 1, 0]. This could therefore be used to store other quantities (e.g., phenotype).

The parents column stores the ids of other individuals that are the parents of an individual. This column is ragged such that an individual can have any number of parents.

The metadata column provides a location for client code to store information about each individual. See the Metadata section for more details on how metadata columns should be used.

Note

The distinction between flags and metadata is that flags holds information about a individual that the library understands, whereas metadata holds information about a individual that the library does not understand. Metadata is for storing auxiliarly information that is not necessary for the core tree sequence algorithms.

#### Edge Table#

An edge defines a parent-child relationship between a pair of nodes over a specific sequence interval. The edge table contains five columns, all of which are mandatory except metadata:

Column

Type

Description

left

double

Left coordinate of the edge (inclusive).

right

double

Right coordinate of the edge (exclusive).

parent

int32

Parent node ID.

child

int32

Child node ID.

metadata

binary

Node Metadata.

Each row in an edge table describes a half-open genomic interval [left, right) over which the child inherited from the given parent. The left and right columns are defined using double precision floating point values. The parent and child columns specify integer IDs in the associated Node Table.

The metadata column provides a location for client code to store information about each edge. See the Metadata section for more details on how metadata columns should be used.

See the Edge requirements section for details on the properties required for a valid set of edges.

#### Site Table#

A site defines a particular location along the genome in which we are interested in observing the allelic state. The site table contains three columns, of which position and ancestral_state are mandatory.

Column

Type

Description

position

double

Position of site in genome coordinates.

ancestral_state

text

The state at the root of the tree.

metadata

binary

Site Metadata.

The position column is a floating point value defining the location of the site in question along the genome.

The ancestral_state column specifies the allelic state at the root of the tree, thus defining the state that nodes inherit if no mutations intervene. The column stores text character data of arbitrary length.

The metadata column provides a location for client code to store information about each site. See the Metadata section for more details on how metadata columns should be used.

See the Site requirements section for details on the properties required for a valid set of sites.

#### Mutation Table#

A mutation defines a change of allelic state on a tree at a particular site. The mutation table contains five columns, of which site, node and derived_state are mandatory.

Column

Type

Description

site

int32

The ID of the site the mutation occurs at.

node

int32

The node this mutation occurs at.

parent

int32

The ID of the parent mutation.

time

double

Time at which the mutation occurred.

derived_state

char

The allelic state resulting from the mutation.

metadata

binary

Mutation Metadata.

The site column is an integer value defining the ID of the site at which this mutation occurred.

The node column is an integer value defining the ID of the first node in the tree below this mutation.

The time column is a double precision floating point value recording how long ago the mutation happened.

The derived_state column specifies the allelic state resulting from the mutation, thus defining the state that the node and any descendant nodes in the subtree inherit unless further mutations occur. The column stores text character data of arbitrary length.

The parent column is an integer value defining the ID of the mutation whose allelic state this mutation replaced. If there is no mutation at the site in question on the path back to root, then this field is set to the null ID (-1). (The parent column is only required in situations where there are multiple mutations at a given site. For “infinite sites” mutations, it can be ignored.)

The metadata column provides a location for client code to store information about each site. See the Metadata section for more details on how metadata columns should be used.

See the Mutation requirements section for details on the properties required for a valid set of mutations.

#### Migration Table#

In simulations, trees can be thought of as spread across space, and it is helpful for inferring demographic history to record this history. Migrations are performed by individual ancestors, but most likely not by an individual whose genome is tracked as a node (as in a discrete-deme model they are unlikely to be both a migrant and a most recent common ancestor). So, tskit records when a segment of ancestry has moved between populations. This table is not required, even if different nodes come from different populations.

Column

Type

Description

left

double

Left coordinate of the migrating segment (inclusive).

right

double

Right coordinate of the migrating segment (exclusive).

node

int32

Node ID.

source

int32

Source population ID.

dest

int32

Destination population ID.

time

double

Time of migration event.

metadata

binary

Migration Metadata.

The left and right columns are floating point values defining the half-open segment of genome affected. The source and dest columns record the IDs of the respective populations. The node column records the ID of the node that was associated with the ancestry segment in question at the time of the migration event. The time column is holds floating point values recording the time of the event.

The metadata column provides a location for client code to store information about each migration. See the Metadata section for more details on how metadata columns should be used.

See the Migration requirements section for details on the properties required for a valid set of mutations.

#### Population Table#

A population defines a grouping of individuals that a node can be said to belong to.

The population table contains one column, metadata.

Column

Type

Description

metadata

binary

Population Metadata.

The metadata column provides a location for client code to store information about each population. See the Metadata section for more details on how metadata columns should be used.

See the Population requirements section for details on the properties required for a valid set of populations.

#### Provenance Table#

Todo

Document the provenance table.

Column

Type

Description

timestamp

char

Timestamp in ISO-8601 format.

record

char

Provenance record.

### Metadata#

Each table (excluding provenance) has a metadata column for storing and passing along information that tskit does not use or interpret. See Metadata for details. The metadata columns are binary columns.

When using the Text file formats, to ensure that metadata can be safely interchanged, each row is base 64 encoded. Thus, binary information can be safely printed and exchanged, but may not be human readable.

The tree sequence itself also has metadata stored as a byte array.

### Valid tree sequence requirements#

Arbitrary data can be stored in tables using the classes in the Tables and Table Collections. However, only a TableCollection that fulfils a set of requirements represents a valid TreeSequence object which can be obtained using the TableCollection.tree_sequence() method. In this section we list these requirements, and explain their rationale. Violations of most of these requirements are detected when the user attempts to load a tree sequence via tskit.load() or TableCollection.tree_sequence(), raising an informative error message. Some more complex requirements may not be detectable at load-time, and errors may not occur until certain operations are attempted. These are documented below.

The Python API also provides tools that can transform a collection of tables into a valid collection of tables, so long as they are logically consistent, see Creating a valid tree sequence.

#### Individual requirements#

Individuals are a basic type in a tree sequence and are not defined with respect to any other tables. Individuals can have a reference to their parent individuals, if present these references must be valid or null (-1).

A valid tree sequence does not require individuals to be sorted in any particular order, and sorting a set of tables using TableCollection.sort() has no effect on individuals. However, individuals can be optionally sorted using TableCollection.sort_individuals().

#### Node requirements#

Given a valid set of individuals and populations, the requirements for each node are:

• population must either be null (-1) or refer to a valid population ID;

• individual must either be null (-1) or refer to a valid individual ID.

An ID refers to a zero-indexed row number in the relevant table, and so is “valid” if is between 0 and one less than the number of rows in the relevant table.

There are no requirements regarding the ordering of nodes with respect to time.

Sorting a set of tables using TableCollection.sort() has no effect on nodes.

#### Edge requirements#

Given a valid set of nodes and a sequence length $$L$$, the simple requirements for each edge are:

• We must have $$0 \leq$$ left $$<$$ right $$\leq L$$;

• parent and child must be valid node IDs;

• time[parent] > time[child];

• edges must be unique (i.e., no duplicate edges are allowed).

The first requirement simply ensures that the interval makes sense. The third requirement ensures that we cannot have loops, since time is always increasing as we ascend the tree.

To ensure a valid tree sequence there is one further requirement:

• The set of intervals on which each node is a child must be disjoint.

This guarantees that we cannot have contradictory edges (i.e., where a node a is a child of both b and c), and ensures that at each point on the sequence we have a well-formed forest of trees.

In the interest of algorithmic efficiency, edges must have the following sortedness properties:

• All edges for a given parent must be contiguous;

• Edges must be listed in nondecreasing order of parent time;

• Within the edges for a given parent, edges must be sorted first by child ID and then by left coordinate.

Violations of these requirements are detected at load time. The TableCollection.sort() method will ensure that these sortedness properties are fulfilled.

#### Site requirements#

Given a valid set of nodes and a sequence length $$L$$, the simple requirements for a valid set of sites are:

• We must have $$0 \leq$$ position $$< L$$;

• position values must be unique.

For simplicity and algorithmic efficiency, sites must also:

• Be sorted in increasing order of position.

Violations of these requirements are detected at load time. The TableCollection.sort() method ensures that sites are sorted according to these criteria.

#### Mutation requirements#

Given a valid set of nodes, edges and sites, the requirements for a valid set of mutations are:

• site must refer to a valid site ID;

• node must refer to a valid node ID;

• time must either be UNKNOWN_TIME (a NAN value which indicates the time is unknown) or be a finite value which is greater or equal to the mutation node’s time, less than the node above the mutation’s time and equal to or less than the time of the parent mutation if this mutation has one. If one mutation on a site has UNKNOWN_TIME then all mutations at that site must, as a mixture of known and unknown is not valid.

• parent must either be the null ID (-1) or a valid mutation ID within the current table

Furthermore,

• If another mutation occurs on the tree above the mutation in question, its ID must be listed as the parent.

For simplicity and algorithmic efficiency, mutations must also:

• be sorted by site ID;

• when there are multiple mutations per site, mutations should be ordered by decreasing time, if known, and parent mutations must occur before their children (i.e. if a mutation with ID $$x$$ has parent with ID $$y$$, then we must have $$y < x$$).

Violations of these sorting requirements are detected at load time. The TableCollection.sort() method ensures that mutations are sorted according site ID, but does not at present enforce that mutations occur after their parent mutations.

Silent mutations (i.e., mutations for which the ancestral and derived states are the same) are allowed. For example, if we have a site with ancestral state of “A” and a single mutation with derived state “A”, then this mutation does not result in any change of state. (This addition was made in release C_0.99.11.)

Note

As tskit.UNKNOWN_TIME is implemented as a NaN value, tests for equality will always fail. Use tskit.is_unknown_time to detect unknown values.

#### Migration requirements#

Given a valid set of nodes and edges, the requirements for a value set of migrations are:

• left and right must lie within the tree sequence coordinate space (i.e., from 0 to sequence_length).

• time must be strictly between the time of its node and the time of any ancestral node from which that node inherits on the segment [left, right).

• The population of any such ancestor matching source, if another migration does not intervene.

To enable efficient processing, migrations must also be:

• Sorted by nondecreasing time value.

Note in particular that there is no requirement that adjacent migration records should be “squashed”. That is, we can have two records m1 and m2 such that m1.right = m2.left and with the node, source, dest and time fields equal. This is because such records will usually represent two independent ancestral segments migrating at the same time, and as such squashing them into a single record would result in a loss of information.

#### Population requirements#

There are no requirements on a population table.

#### Provenance requirements#

The timestamp column of a provenance table should be in ISO-8601 format.

The record should be valid JSON with structure defined in the Provenance Schema section (TODO).

### Table indexes#

To efficiently iterate over the trees in a tree sequence, tskit uses indexes built on the edges. To create a tree sequence from a table collection the tables must be indexed; the TableCollection.build_index() method can be used to create an index on a table collection if necessary.

Todo

Add more details on what the indexes actually are.

## Data encoding#

In this section we describe the high-level details of how data is encoded in tables. Tables store data in a columnar manner. In memory, each table is organised as a number of blocks of contiguous storage, one for each column. There are many advantages to this approach, but the key property for us is that allows for very efficient transfer of data in and out of tables. Rather than inserting data into tables row-by-row (which can be done in Python using the add_row methods), it is much more efficient to add many rows at the same time by providing pointers to blocks of contiguous memory. By taking this approach, we can work with tables containing gigabytes of data very efficiently.

For instance, in the Python API we can use the numpy Array API to allow us to define and work with numeric arrays of the required types. Node IDs, for example, are defined using 32 bit integers. Thus, the parent column of an Edge Table’s with n rows is a block 4n bytes.

This approach is very straightforward for columns in which each row contains a fixed number of values. However, dealing with columns containing a variable number of values is more problematic.

### Encoding ragged columns#

A ragged column is a column in which the rows are not of a fixed length. For example, Metadata columns contain binary of data of arbitrary length. To encode such columns in the tables API, we store two columns: one contains the flattened array of data and another stores the offsets of each row into this flattened array. Consider an example:

import tskit

s = tskit.SiteTable()
s.add_row(0, "A")
s.add_row(0, "")
s.add_row(0, "TTT")
s.add_row(0, "G")
s

idpositionancestral_statemetadata
00A
10
20TTT
30G

In this example we create a Site Table with four rows, and then display this table. We can see that the second row has the empty string as its ancestral_state, and the third row’s ancestral_state is TTT. Now let’s print out the columns:

print("Ancestral state (numerical): ", s.ancestral_state)
print("Ancestral state (as bytes): ", s.ancestral_state.tobytes())
print("Ancestral state offsets: ", s.ancestral_state_offset)

Ancestral state (numerical):  [65 84 84 84 71]
Ancestral state (as bytes):  b'ATTTG'
Ancestral state offsets:  [0 1 1 4 5]


When we print out the tables ancestral_state column, we see that its a numpy array of length 5: this is the flattened array of ASCII encoded values for these rows. When we decode these bytes using the numpy tobytes method, we get the string ‘ATTTG’. This flattened array can now be transferred efficiently in memory like any other column We then use the ancestral_state_offset column to allow us find the individual rows. For a row j:

ancestral_state[ancestral_state_offset[j]: ancestral_state_offset[j + 1]]


gives us the array of bytes for the ancestral state in that row. For example, here is row 2:

s.ancestral_state[s.ancestral_state_offset[2]: s.ancestral_state_offset[3]].tobytes()

b'TTT'


For a table with n rows, any offset column must have n + 1 values, the first of which is always 0. The values in this column must be nondecreasing, and cannot exceed the length of the ragged column in question.

## Reference sequence#

Along with the topology and site information stored in the tskit tree sequence, we can also optionally store an associated reference sequence. Reference sequences are flexible, and can consist simply of some metadata recording which assembly build a tree sequence uses, or storing the entire sequence itself.

Warning

Reference sequence support in tskit is preliminary. Reference sequence data can be stored and accessed via the C API. Support in the Python API is limited to usage in TreeSequence.alignments() and related methods, where it provides the default values for nucleotide positions between sites.

## Tree structure#

### Quintuply linked trees#

Tree structure in tskit is encoded internally as a “quintuply linked tree”, a generalisation of the triply linked tree encoding used by Knuth and others. Nodes are represented by their integer IDs, and their relationships to other nodes are recorded in the parent, left_child, right_child, left_sib and right_sib arrays. For example, consider the following tree and its associated arrays:

import io

import tskit
from IPython.display import SVG

nodes = """\
id      is_sample   time
0       1           0
1       1           0
2       1           0
3       1           0
4       1           0
5       0           1
6       0           2
7       0           3
"""
edges = """\
left    right   parent  child
0       60      5       4,3
0       40      6       2
0       60      6       1,0
20      40      6       5
0       20      7       5
40      60      7       5
0       60      7       6
40      60      7       2
"""
ts = tskit.load_text(
nodes=io.StringIO(nodes), edges=io.StringIO(edges), strict=False
)

SVG(ts.first().draw_svg(time_scale="rank"))

from IPython.display import HTML

def html_quintuple_table(ts, show_virtual_root=False, show_convenience_arrays=False):
tree = ts.first()
columns = ["node", "parent", "left_child", "right_child", "left_sib", "right_sib"]
convenience_arrays = ["num_children", "edge"]
if show_convenience_arrays:
columns += convenience_arrays
data = {k:[] for k in columns}
for u in sorted(tree.nodes(tree.virtual_root if show_virtual_root else None)):
for colname in columns:
data[colname].append(u if colname == "node" else getattr(tree, colname)(u))
html = "<tr>"
for colname in columns:
html += f"<th>{colname}</th>"
html += "</tr>"
for u in range(len(data["node"])):
html += "<tr>" if u < ts.num_nodes else "<tr style='font-style: italic; color:red'>"
for colname in columns:
html += f"<td>{data[colname][u]}</td>"
html += "</tr>"
return "<table>" + html + "</table>"

HTML(html_quintuple_table(ts))

nodeparentleft_childright_childleft_sibright_sib
06-1-1-11
16-1-102
26-1-11-1
35-1-1-14
45-1-13-1
5734-16
67025-1
7-156-1-1

Each node in the tree corresponds to a row in this table, and the columns are the individual arrays recording the quintuply linked structure. Thus, we can see that the parent of nodes 0, 1, and 2 is 6. Similarly, the left child of 6 is 0 and the right child of 6 is 2. The left_sib and right_sib arrays then record each nodes sibling on its left or right, respectively; hence the right sib of 0 is 1, and the right sib of 1 is 2. Thus, sibling information allows us to efficiently support trees with arbitrary numbers of children. In each of the five pointer arrays, the null node (-1) is used to indicate the end of a path; thus, for example, the parent of 7 and left sib of 0 are null.

Please see this example for details of how to use the quintuply linked structure in the C API.

Note

For many applications we do not need the quintuply linked trees, and (for example) the left_sib and right_child arrays can be ignored. The reason for using a quintuply instead of triply linked encoding is that it is not possible to efficiently update the trees as we move along the sequence without the quintuply linked structure.

Warning

The left-to-right ordering of nodes is determined by the order in which edges are inserted into the tree during iteration along the sequence. Thus, if we arrive at the same tree by iterating from different directions, the left-to-right ordering of nodes may be different! The specific ordering of the children of a node should therefore not be depended on.

### Convenience arrays#

Similar to the five arrays representing the quintuply linked tree, convenience arrays track information on each node in the tree. These arrays are not essential to represent the trees within a tree sequence. However, they can be useful for specific algorithms (e.g. when computing tree (im)balance metrics). Two convenience arrays have been implemented so far: Tree.num_children_array and Tree.edge_array.

Here is the table above with the convenience arrays also shown:

HTML(html_quintuple_table(ts, show_convenience_arrays=True))

nodeparentleft_childright_childleft_sibright_sibnum_childrenedge
06-1-1-1102
16-1-10203
26-1-11-104
35-1-1-1400
45-1-13-101
5734-1627
67025-139
7-156-1-12-1

### Roots#

In the tskit trees we have shown so far, all the sample nodes have been connected to each other. This means each tree has only a single root (i.e. the oldest node found when tracing a path backwards in time from any sample). However, a tree can contain Isolated sample nodes or unconnected topologies, and can therefore have multiple roots. Here’s an example, created by deleting the edge joining 6 and 7 in the tree sequence used above:

tables = ts.dump_tables()
tables.edges.truncate(ts.num_edges - 1)
ts_multiroot = tables.tree_sequence()
SVG(ts_multiroot.first().draw_svg(time_scale="rank"))


In tskit terminology, this should not be thought of as two separate trees, but as a single multi-root “tree”, comprising two unlinked topologies. This fits with the definition of a tree in a tree sequence: a tree describes the ancestry of the same fixed set of sample nodes at a single position in the genome. In the picture above, both the left and right hand topologies are required to describe the genealogy of samples 0..4 at this position.

Here’s what the entire tree sequence now looks like:

SVG(ts_multiroot.draw_svg(time_scale="rank"))


From the terminology above, it can be seen that this tree sequence consists of only three trees (not five). The first tree, which applies from position 0 to 20, is the one used in our example. As we saw, removing the edge connecting node 6 to node 7 has created a tree with 2 roots (and thus 2 unconnected topologies in a single tree). In contrast, the second tree, from position 20 to 40, has a single root. Finally the third tree, from position 40 to 60, again has two roots.

#### The root threshold#

The roots of a tree are defined by reference to the sample nodes. By default, roots are the unique endpoints of the paths traced upwards from the sample nodes; equivalently, each root counts one or more samples among its descendants (or is itself a sample node). This is the case when the root_threshold property of a tree is left at its default value of 1. If, however, the root_threshold is (say) 2, then a node is considered a root only if it counts at least two samples among its descendants. Setting an alternative root_threshold value can be used to avoid visiting Isolated sample nodes, for example when dealing with trees containing Missing data.

#### The virtual root#

To access all the roots in a tree, tskit uses a special additional node called the virtual root. This is primarily a bookkeeping device, and can normally be ignored: it is not plotted in any visualizations and does not exist as an independent node in the node table. However, the virtual root can be useful in certain algorithms because its children are defined as all the “real” roots in a tree. Hence by descending downwards from the virtual root, it is possible to access the entire genealogy at a given site, even in a multi-root tree. In the quintuply linked tree encoding, the virtual root appears as an extra element at the end of each of the tree arrays. Here’s the same table as before but with the virtual root also shown, using red italics to emphasise that it is not a “real” node:

HTML(html_quintuple_table(ts_multiroot, show_virtual_root=True))

nodeparentleft_childright_childleft_sibright_sib
06-1-1-11
16-1-102
26-1-11-1
35-1-1-14
45-1-13-1
5734-1-1
6-102-17
7-1556-1
8-167-1-1

You can see that the virtual root (node 8) has 6 as its left child and 7 as its right child. Importantly, though, this is an asymmetric relationship: the parent of the “real” roots 6 and 7 is null (-1) and not the virtual root. Hence when we ascend up the tree from the sample nodes to their parents, we stop at the “real” roots, and never encounter the virtual root.

Because the virtual root can be useful in some algorithms, it can optionally be returned in traversal orders (see Tree.nodes()). The following properties apply:

• All trees in a tree sequence share the same virtual root.

• The virtual root’s ID is always equal to the number of nodes in the tree sequence (i.e. the length of the node table). However, there is no corresponding row in the node table, and any attempts to access information about the virtual root via either the tree sequence or tables APIs will fail with an out-of-bounds error.

• The parent and siblings of the virtual root are null.

• The time of the virtual root is defined as positive infinity (if accessed via Tree.time()). This is useful in defining the time-based node traversal orderings.

• The virtual root is the parent of no other node—roots do not have parent pointers to the virtual root.

### Isolated nodes#

In a tree, it is possible for a node to have no children and no parent. Such a node is said to be isolated, meaning that we don’t know anything about its relationships over a specific genomic interval. This is commonly true for ancestral genomes, which often have large regions that have not been inherited by any of the sample nodes in the tree sequence, and therefore regions about which we know nothing. This is true, for example, of node 7 in the middle tree of our previous example, which is why it is not plotted on that tree:

display(SVG(ts_multiroot.draw_svg(time_scale="rank")))
for tree in ts_multiroot.trees():
print(
"Node 7",
"is" if tree.is_isolated(7) else "is not",
"isolated from position",
tree.interval.left,
"to",
tree.interval.right,
)

Node 7 is not isolated from position 0.0 to 20.0
Node 7 is isolated from position 20.0 to 40.0
Node 7 is not isolated from position 40.0 to 60.0


#### Isolated sample nodes#

It is also possible for a sample node to be isolated. As long as the root threshold is set to its default value, an isolated sample node will count as a root, and therefore be considered as being present on the tree (meaning it will be returned by the Tree.nodes() and Tree.samples() methods). When displaying a tree, isolated samples are shown unconnected to other nodes. To illustrate, we can remove the edge from node 2 to node 7:

tables = ts_multiroot.dump_tables()
tables.edges.set_columns(
**tables.edges[(tables.edges.parent != 7) | (tables.edges.child != 2)].asdict())
ts_isolated = tables.tree_sequence()
SVG(ts_isolated.draw_svg(time_scale="rank"))


The rightmost tree now contains an isolated sample node (node 2), which counts as one of the Roots of the tree. This tree therefore has three roots, one of which is node 2:

rightmost_tree = ts_isolated.at_index(-1)
print(rightmost_tree.num_roots, "roots in the rightmost tree, with IDs", rightmost_tree.roots)
print(
"IDs of isolated samples in this tree:",
[u for u in rightmost_tree.samples() if rightmost_tree.is_isolated(u)],
)

3 roots in the rightmost tree, with IDs [6, 2, 7]
IDs of isolated samples in this tree: [2]


In tskit, isolated sample nodes are closely associated with the encoding of Missing data.

### Dead leaves and branches#

In a tskit tree, a leaf node is defined as a node without any children. The implications of this turn out to be slighly unintuitive, and so are worth briefly documenting here. Firstly, the same node can be a leaf in one tree, and not a leaf in the next tree along the tree sequence. Secondly all isolated nodes must be leaves (as by definition they have no children). Thirdly sample nodes need not be leaves (they could be “internal samples”); likewise leaf nodes need not be samples.

Node 7 in the example above provides a good case study. Note that it is a root node with at least one child (i.e. not a leaf) in trees 0 and 2; in contrast in tree 1 it is isolated. Strictly, because it is isolated in tree 1, it is also a leaf node there, although it is not attached to a root, not a sample, and is therefore not plotted. In this case, in that tree we can think of node 7 as a “dead leaf” (and we don’t normally plot dead leaves). In fact, in a large tree sequence of many trees, most ancestral nodes will be isolated in any given tree, and therefore most nodes in such a tree will be of this sort. However, these dead leaves are excluded from most calculations on trees, because algorithms usually traverse the tree by starting at a root and working down, or by starting at a sample and working up. Hence when we refer to the leaves of a tree, it is usually shorthand for the leaves on the tree (that is, attached via branches, to one of the the tree roots). Dead leaves are excluded from this definition.

Note that it is also possible to have trees in which there are “dead branches”: that is sections of topology which are not accessible from a root, and whose tips are all dead leaves. Although valid, this is a relatively unusual state of affairs, and such branches are not plotted by the standard Visualization methods. The Tree.nodes() method will not, by default, traverse through dead branches, although it can be made to do so by specifying the ID of a dead node as the root for traversal.

## Encoding genetic variation#

Genetic variation is incorporated into a tree sequence by placing mutations at sites along the genome. The genotypes of the different samples at each site can be found by using the tree to calculate which mutations are inherited by the different samples. This is the fundamental basis of how tree sequences efficiently encode DNA sequences, and is explained in depth elsewhere (e.g. in the tutorials).

Below, we discuss some implications of this encoding in more detail, in particular the way in which it can be used to model missing data.

### Missing data#

If, at a particular genomic position, a node is isolated and additionally has no mutations directly above it, its genotype at that position is considered to be unknown (however, if there is a mutation above an isolated node, it can be thought of as saying directly what the genotype is, and so renders the genotype at that position not missing).

By way of illustration, we’ll use the delete_intervals() method to remove all knowledge of the ancestry in the middle portion of the previous example (say from position 15 to 45) sprinkle on some mutations, and make sure there are sites at every position:

import numpy as np
import msprime

tables = msprime.sim_mutations(ts_isolated, rate=0.1, random_seed=123).dump_tables()
tables.delete_intervals([[15, 45]], simplify=False)
missing_sites = np.setdiff1d(np.arange(tables.sequence_length), tables.sites.position)
for pos in missing_sites:
tables.sites.add_row(position=pos, ancestral_state="A")  # Add sites at every pos
tables.sort()
missing_ts = tables.tree_sequence()
SVG(missing_ts.draw_svg())


The middle section of the genome now has no ancestry at all, and therefore for any site that is in this region, the genotypic state that it is assigned is a special value tskit.MISSING_DATA, or -1. The haplotypes() method, which outputs the actual allelic state for each sample, defaults to outputting an N at these sites. Therefore where any sample node is isolated, the haplotype will show an N, indicating the DNA sequence is unknown. This will be so not only in the middle of all of the sample genomes, but also at the right hand end of the genome of sample 2, as it is an isolated sample node in the rightmost tree:

for i, h in enumerate(missing_ts.haplotypes()):
print(f"Sample {i}: {h}")

Sample 0: AAAGCCAATGAAATANNNNNNNNNNNNNNNNNNNNNNNNNNNNNNGGAACGAATTGAACT
Sample 1: AGAGACAATAGAATANNNNNNNNNNNNNNNNNNNNNNNNNNNNNNTAAATAACCTGAACT
Sample 2: AGAGACGACAGAATANNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN
Sample 3: AGATACGATAGCAAANNNNNNNNNNNNNNNNNNNNNNNNNNNNNNTCTACGAACAAAAAC
Sample 4: AGAGATGATAGCAAANNNNNNNNNNNNNNNNNNNNNNNNNNNNNNTCTACGAACAAAACC


See the TreeSequence.variants() method and Variant class for more information on how missing data is represented in variant data.