# Analysing trees#

There are a number of different ways we might want to analyse a single Tree. Most involve some sort of traversal over the nodes, mutations, or branches in the tree. tskit provides various way of traversing through a tree, and also some built in phylogenetic algorithms such as Tree.map_mutations() which efficently places mutations (“characters” in phylogenetic terminology) on a given tree.

## Tree traversals#

Given a single Tree, traversals in various orders are possible using the nodes() iterator. Take the following tree:

import tskit

tree = ts.first()
tree.draw_svg() We can visit the nodes in different orders by providing an order parameter to the Tree.nodes() iterator:

for order in ["preorder", "inorder", "postorder"]:
print(f"{order}:\t", list(tree.nodes(order=order)))

preorder:	 [7, 5, 0, 1, 2, 6, 3, 4]
inorder:	 [0, 5, 1, 2, 7, 3, 6, 4]
postorder:	 [0, 1, 2, 5, 3, 4, 6, 7]


Much of the time, the specific ordering of the nodes is not important and we can leave it out (defaulting to preorder traversal). For example, here we compute the total branch length of a tree:

total_branch_length = sum(tree.branch_length(u) for u in tree.nodes())
print(f"Total branch length: {total_branch_length}")

Total branch length: 10.0


Note that this is also available as the Tree.total_branch_length attribute.

### Traversing upwards#

For many applications it is useful to be able to traverse upwards from the leaves. We can do this using the Tree.parent() method, which returns the parent of a node. For example, we can traverse upwards from each of the samples in the tree:

for u in tree.samples():
path = []
v = u
while v != tskit.NULL:
path.append(v)
v = tree.parent(v)
print(u, "->", path)

0 -> [0, 5, 7]
1 -> [1, 5, 7]
2 -> [2, 5, 7]
3 -> [3, 6, 7]
4 -> [4, 6, 7]


### Traversals with information#

Sometimes we will need to traverse down the tree while maintaining some information about the nodes that are above it. While this can be done using recursive algorithms, it is often more convenient and efficient to use an iterative approach. Here, for example, we define an iterator that yields all nodes in preorder along with their path length to root:

def preorder_dist(tree):
for root in tree.roots:
stack = [(root, 0)]
while len(stack) > 0:
u, distance = stack.pop()
yield u, distance
for v in tree.children(u):
stack.append((v, distance + 1))

print(list(preorder_dist(tree)))

[(7, 0), (6, 1), (4, 2), (3, 2), (5, 1), (2, 2), (1, 2), (0, 2)]


## Networkx#

Traversals and other network analysis can also be performed using the sizeable networkx library. This can be achieved by calling Tree.as_dict_of_dicts() to convert a Tree instance to a format that can be imported by networkx to create a graph:

import networkx as nx

g = nx.DiGraph(tree.as_dict_of_dicts())
print(sorted(g.edges))

[(5, 0), (5, 1), (5, 2), (6, 3), (6, 4), (7, 5), (7, 6)]


### Traversing upwards in networkx#

We can revisit the above examples and traverse upwards with networkx using a depth-first search algorithm:

g = nx.DiGraph(tree.as_dict_of_dicts())
for u in tree.samples():
path = [u] + [parent for parent, child, _ in
nx.edge_dfs(g, source=u, orientation="reverse")]
print(u, "->", path)

0 -> [0, 5, 7]
1 -> [1, 5, 7]
2 -> [2, 5, 7]
3 -> [3, 6, 7]
4 -> [4, 6, 7]


### Calculating distances to the root#

Similarly, we can yield the nodes of a tree along with their distance to the root in pre-order in networkx as well. Running this on the example above gives us the same result as before:

g = nx.DiGraph(tree.as_dict_of_dicts())
for root in tree.roots:
print(sorted(list(nx.shortest_path_length(g, source=root).items())))

[(0, 2), (1, 2), (2, 2), (3, 2), (4, 2), (5, 1), (6, 1), (7, 0)]


### Finding nearest neighbors#

If some samples in a tree are not at time 0, then finding the nearest neighbor of a sample is a bit more involved. Instead of writing our own traversal code we can again draw on a networkx algorithm. Let us start with an example tree with three samples that were sampled at different time points:

ts = tskit.load("data/different_time_samples.trees")
tree = ts.first()
tree.draw_svg(y_axis=True, time_scale="rank") The generation times for these nodes are as follows:

for u in tree.nodes():
print(f"Node {u}: time {tree.time(u)}")

Node 4: time 20.00539877826333
Node 2: time 20.0
Node 3: time 17.833492457579652
Node 0: time 0.0
Node 1: time 1.0


Note that samples 0 and 1 are about 35 generations apart from each other even though they were sampled at almost the same time. This is why samples 0 and 1 are closer to sample 2 than to each other.

For this nearest neighbor search we will be traversing up and down the tree, so it is easier to treat the tree as an undirected graph:

g = nx.Graph(tree.as_dict_of_dicts())


When converting the tree to a networkx graph the edges are annotated with their branch length:

for e in g.edges(data=True):
print(e)

(4, 2, {'branch_length': 0.0053987782633306836})
(4, 3, {'branch_length': 2.1719063206836786})
(3, 0, {'branch_length': 17.833492457579652})
(3, 1, {'branch_length': 16.833492457579652})


We can now use the “branch_length” field as a weight for a weighted shortest path search:

import collections
import itertools

# a dictionary of dictionaries to represent our distance matrix
dist_dod = collections.defaultdict(dict)
for source, target in itertools.combinations(tree.samples(), 2):
dist_dod[source][target] = nx.shortest_path_length(
g, source=source, target=target, weight="branch_length"
)
dist_dod[target][source] = dist_dod[source][target]

# extract the nearest neighbor of nodes 0, 1, and 2
nearest_neighbor_of = [min(dist_dod[u], key=dist_dod[u].get) for u in range(3)]

print(dict(zip(range(3), nearest_neighbor_of)))

{0: 2, 1: 2, 2: 1}


## Parsimony#

Take a site on the following tree with three allelic states, where the samples are coloured by the allele they possess, but where we don’t know the position of the mutations that caused this variation:

tree = tskit.load("data/parsimony_simple.trees").first()
alleles = ["red", "blue", "green"]
genotypes = [0, 0, 0, 0, 1, 2]
styles = [f".n{j} > .sym {{fill: {alleles[g]}}}" for j, g in enumerate(genotypes)]
tree.draw_svg(style="".join(styles)) The Tree.map_mutations() method finds a parsimonious explanation for a set of discrete character observations on the samples in a tree using classical phylogenetic algorithms:

ancestral_state, mutations = tree.map_mutations(genotypes, alleles)
print("Ancestral state = ", ancestral_state)
for mut in mutations:
print(f"Mutation: node = {mut.node} derived_state = {mut.derived_state}")

Ancestral state =  red
Mutation: node = 4 derived_state = blue
Mutation: node = 5 derived_state = green


In this case, the algorithm has concluded, quite reasonably, that the most parsimonious description of this state is that the ancestral state is red and there was a mutation to blue and green over nodes 4 and 5.

### Building tables#

Below we show how a set of tables can be updated using the Tables API, taking advantage of the Tree.map_mutations() method to identify parsimonious positions for mutations on a tree. Here’s the tree we’ll use:

import pickle
ts.draw_svg(size=(500, 300), time_scale="rank") Now we can modify the tables by adding mutations. To find the location of mutations, we infer them from some observed data (some site positions with associated genotypes and allelic states, in the conventional tskit encoding):

with open("data/parsimony_map.pickle", "rb") as file:
print("Variant data: each site has a position, allele list, and genotypes array:")
for i, v in enumerate(data):
print(f"Site {i} (pos {v['pos']:7.4f}): alleles: {v['alleles']}, genotypes: {v['genotypes']}")
print()
tree = ts.first()  # there's only one tree anyway
tables = ts.dump_tables()
# Infer the sites and mutations from the variants.
for variant in data:
ancestral_state, mutations = tree.map_mutations(variant["genotypes"], variant['alleles'])
info = f"Site {site_id}: parsimony sets ancestral state to {ancestral_state}"
parent_offset = len(tables.mutations)
for mut in mutations:
parent = mut.parent
if parent != tskit.NULL:
parent += parent_offset
site_id, node=mut.node, parent=parent, derived_state=mut.derived_state)
info += f", and places mutation {mut_id} to {mut.derived_state} above node {mut.node}"
print(info)
new_ts = tables.tree_sequence()

Variant data: each site has a position, allele list, and genotypes array:
Site 0 (pos  8.3726): alleles: ('G', 'A'), genotypes: [1 1 0 0 0 0]
Site 1 (pos 24.4759): alleles: ('T', 'C'), genotypes: [0 0 0 0 1 1]
Site 2 (pos 34.3178): alleles: ('G', 'T'), genotypes: [0 0 1 1 0 0]
Site 3 (pos 39.2118): alleles: ('G', 'C'), genotypes: [0 0 1 1 0 0]
Site 4 (pos 44.0257): alleles: ('G', 'C'), genotypes: [1 1 0 0 0 0]
Site 5 (pos 48.0932): alleles: ('C', 'G'), genotypes: [0 0 1 1 0 0]
Site 6 (pos 68.4830): alleles: ('C', 'G'), genotypes: [0 0 1 1 0 0]
Site 7 (pos 69.4755): alleles: ('A', 'C'), genotypes: [0 0 0 0 1 1]
Site 8 (pos 71.2330): alleles: ('C', 'T'), genotypes: [1 1 0 0 0 0]
Site 9 (pos 71.9150): alleles: ('G', 'T'), genotypes: [0 0 0 0 0 1]

Site 0: parsimony sets ancestral state to G, and places mutation 0 to A above node 6
Site 1: parsimony sets ancestral state to T, and places mutation 1 to C above node 8
Site 2: parsimony sets ancestral state to G, and places mutation 2 to T above node 7
Site 3: parsimony sets ancestral state to G, and places mutation 3 to C above node 7
Site 4: parsimony sets ancestral state to G, and places mutation 4 to C above node 6
Site 5: parsimony sets ancestral state to C, and places mutation 5 to G above node 7
Site 6: parsimony sets ancestral state to C, and places mutation 6 to G above node 7
Site 7: parsimony sets ancestral state to A, and places mutation 7 to C above node 8
Site 8: parsimony sets ancestral state to C, and places mutation 8 to T above node 6
Site 9: parsimony sets ancestral state to G, and places mutation 9 to T above node 5


And here are the parsimoniously positioned mutations on the tree

mut_labels = {}  # An array of labels for the mutations
for mut in new_ts.mutations():  # Make pretty labels showing the change in state
site = new_ts.site(mut.site)
older_mut = mut.parent >= 0  # is there an older mutation at the same position?
prev = new_ts.mutation(mut.parent).derived_state if older_mut else site.ancestral_state
mut_labels[site.id] = f"{mut.id}: {prev}→{mut.derived_state}"

new_ts.draw_svg(size=(500, 300), mutation_labels=mut_labels, time_scale="rank") ### Parsimony and missing data#

We can also take missing data into account when finding a set of parsimonious state transitions. We do this by specifying the special value tskit.MISSING_DATA (-1) as the state, which is treated by the algorithm as “could be anything”.

For example, here we state that sample 0 is missing, indicated by the colour white:

tree = tskit.load("data/parsimony_simple.trees").first()
alleles = ["red", "blue", "green", "white"]
genotypes = [tskit.MISSING_DATA, 0, 0, 0, 1, 2]
styles = [f".n{j} > .sym {{fill: {alleles[g]}}}" for j, g in enumerate(genotypes)]
tree.draw_svg(style="".join(styles)) Now we run the Tree.map_mutations() method, which applies the Hartigan parsimony algorithm:

ancestral_state, mutations = tree.map_mutations(genotypes, alleles)
print("Ancestral state = ", ancestral_state)
for mut in mutations:
print(f"Mutation: node = {mut.node} derived_state = {mut.derived_state}")

Ancestral state =  red
Mutation: node = 4 derived_state = blue
Mutation: node = 5 derived_state = green


The algorithm decided, again, quite reasonably, that the most parsimonious explanation for the input data is the same as before. Thus, if we used this information to fill out mutation table as above, we would impute the missing value for 0 as red.

The output of the algorithm can be a little surprising at times. Consider this example::

tree = msprime.simulate(6, random_seed=42).first()
alleles = ["red", "blue", "white"]
genotypes = [1, -1, 0, 0, 0, 0]
node_colours = {j: alleles[g] for j, g in enumerate(genotypes)}
ancestral_state, mutations = tree.map_mutations(genotypes, alleles)
print("Ancestral state = ", ancestral_state)
for mut in mutations:
print(f"Mutation: node = {mut.node} derived_state = {mut.derived_state}")
tree.draw(node_colours=node_colours)

Ancestral state =  red
Mutation: node = 6 derived_state = blue Note that this is putting a mutation to blue over node 6, not node 0 as we might expect. Thus, we impute here that node 1 is blue. It is important to remember that the algorithm is minimising the number of state transitions; this may not correspond always to what we might consider the most parsimonious explanation.

## Fast tree-based algorithms using numba#

Todo

Add a few examples here of how to use numba to speed up tree based dynamic programming algorithms. There are a good number of worked-up examples with timings in issue #63