# Bipartitions¶

Many tree statistics and operations in DendroPy use the bipartition encoding of a Tree instance in the background, including, for example:

• tree statistics and metrics
• tree comparisons
• tree scoring

By default, the DendroPy functions assume that bipartitions are not encoded, or are not up-to-date with respect to the current tree structure, resulting in their recalculation every time. This is computationally inefficient, and you want to avoid it if, indeed, the bipartition encoding of a tree is current. You can control whether or not these service functions recalculate the bipartition encoding by passing in the argument is_bipartitions_updated=True to suppress the recalculation or is_bipartitions_updated=False to force it.

If you are doing multiple operations that require a bipartition encoding, you should call Tree.encode_bipartitions() once for each tree, and, then, as long as the trees are not modified since the encoding, specify the is_bipartitions_updated=True argument to each of the functions that use it to ensure that the bipartitions are not recalculated each time.

If, on the other hand, you modify a tree structure in any way, e.g., rerooting, pruning, add/removing nodes or subtrees, you should update the bipartition encoding of a tree yourself by calling Tree.encode_bipartitions(), or make sure to specify is_bipartitions_updated=False to the first function that you call following the tree modification.

## Modeling Bipartitions¶

### A Bipartition is a Partitioning of Taxa Corresponding to an Edge of a Tree¶

A bipartition is the division or sorting of the leaves/tips of a tree into two mutually-exclusive and collectively-exhaustive subsets (i.e., a partition, in the set theory sense, of the leaves of the tree into exactly two non-empty subsets; hence the term, “bi-partition”). Every edge on a tree corresponds to a bipartition in the sense that if were were to split or bisect a tree at a particular edge, the leaf sets of each of the two new trees constitute the a bipartition of the leaf set of the original tree. In the context of evolutionary trees like a phylogeny, the leaves typically are associated with operational taxonomic unit concepts, or, for short, taxa. So, just as we view a tree as a schematic representation of the relationships of taxa, we can see bipartitions as a representation of a clustering of taxa.

For example, given a tree:

((a,(b,c)),(d,(e,f)));

the edge subtending the leaf node with taxon “d” corresponds to the bipartition that splits “d” from the rest of the taxa. Similarly, the edge subtending the most-recent common ancestor (MRCA) node of taxa “d”, “e”, and “f” corresponds to the bipartition that splits “d”, “e”, and “f” from the rest of the taxa, “a”, “b”, and “c”.

### A Bipartition Can Be Described by a Bitmask¶

If we were to index the taxa of the tree, with the first taxon getting index 1, the second index 2, the third index 3, etc. and so on until index $n$, we can represent any possible split as sequence of symbols, such as:

abbabbaa

where the symbol indicates membership in one arbitrarily-labeled group (e.g., “a”) or the other (e.g., “b”) of a particular taxon, based on how we relate the taxon indexes to the position of the symbols in sequence.

If we were to use a left-to-right order, such that the first element corresponded to the first taxon, the second to the second taxon, and so one, the above sequence would describe the a partition of the taxa {1,2,...,8} into the sets {1,4,7,8} and {2,3,5,6}. However, in DendroPy, we use a right-to-left order (for reasons explained below), such that the right-most element corresponds to the taxon with index 1, the next right-most element corresponds to the taxon with index 2 and so on, so the sequence above represents a partition of the taxa {1,2,...,8} into the sets {1,2,5,8} and {3,4,6,7}.

Let us say that we had a set of 8 taxa {A,B,...,H}:

A, B, C, D, E, F, G, H

 Taxon A B C D E F G H Index 1 2 3 4 5 6 7 8

Then we can describe a bipartition that divides the taxa into two groups {A,B,E,H} and {C,D,F,G}, using right-to-left ordering and symbols “0” and “1” (instead of “a” and “b”) as:

 Taxon Index 8 7 6 5 4 3 2 1 Group 0 1 1 0 1 1 0 0

We can succintly and usefully represent the bipartition above with an integer given by interpreting the sequence of 0’s and 1’s as bits. Interpreting the sequence above, “01101100”, as a binary number or bitmask means that this bipartition can be represented as the decimal integer “108”.

This, in essence, is how bipartitions are represented in DendroPy: as integers that are interpreted as bitmasks (also known as bit arrays, bit vectors, or bit fields, though exact application of terminology varies depending on whether or not primary operations are bitwise or dereferencing through offset indexes or field names, etc.), where the 0’s and 1’s assign taxa to different subsets of the bipartition.

As an example, consider the following tree:

[&R] (A,(B,(C,(D,E))));


This would be encoded as:

/----------------------------- 00001 (A)
11111
|      /---------------------- 00010 (B)
\------11110
|       /-------------- 00100 (C)
\-------11100
|      /------- 01000 (D)
\------11000
\------- 10000 (E)


The leaves are assigned bitmasks based on the indexes of the taxa, while the internal nodes are given by a bitwise OR-ing of the bitmasks of their children.

In DendroPy, bipartitions are modeled using bitmasks as discussed above, i.e., integers that, when represented as a bitarray or bitstring, specify the assignment of taxa into one of two groups, based on whether or not the bit corresponding to the taxon index is set or not.

In fact, each bipartition is actually modeled by two types of bitmasks: a leafset bitmask and a split bitmask:

• A leafset bitmask is a bit array in which the presence of a taxon in the leaves descending from the edge associated with the bipartition is represented by a set bit (“1”), while its absence is represented by an unset bit (“0”). The taxa are mapped to bit positions using a least-significant bit mapping scheme, in which the first taxon is represented by the least significant bit, the second taxon is represented by the next most significant bit, and so on.

• A split bitmask is a bit array which divides or partitions taxa by assign each taxon to one of two arbitrarily-labeled groups, “0” or “1”, depending on whether or not a bit is set or not in the position corresponding the taxon index under a least-signficant bit mapping scheme as described above.

• For bipartitions of rooted trees, the split bitmask is the same value as the leafset bitmask.
• For bipartitions of unrooted trees, the split bitmask is the same value as the leafset bitmask if and only if the least-signficant bit of the leafset bitmask is 0 (i.e., the first taxon is assigned to group “0”), or the complement of the leafset bitmask if this is the case. In other words, with unrooted trees we constrain the split bitmasks such that the first taxon and all other taxa grouped together with it are always placed in group “0”.

Why this complication?

Consider the following unrooted tree:

A    C    D
\   |   /
+--+--+
/       \
B         E


This could be represented by either of the following NEWICK strings:

[&U] ((A,B),(C,(D,E)));
[&U] (((A,B),C),(D,E));


Both the above topologies, while distinct if interpreted as rooted, represent identical unrooted toplogies.

When the bipartitions are encoded as leafset bitmasks, we get the following if the first tree statement is parsed by DendroPy:

                    /--------- 00001 (A)
/-------------------00011
|                   \--------- 00010 (B)
11111
|         /------------------- 00100 (C)
\---------11100
|         /--------- 01000 (D)
\---------11000
\--------- 10000 (E)


and the following if the second tree statement is parsed by DendroPy:

                    /--------- 00001 (A)
/---------00011
/---------00111     \--------- 00010 (B)
|         |
11111     \------------------- 00100 (C)
|
|                   /--------- 01000 (D)
\-------------------11000
\--------- 10000 (E)


Note that the leafset bitmask “11100” in the first tree is absent in the second tree, while conversely, the leafset bitmask “00111” in the second tree is absent in the first tree.

This difference is due purely to the placement of the root to one side or the other of taxon ‘C’. In rooted trees, this root is a real root, and this difference in bipartitions as given by the leafset bitmasks is also real. In unrooted trees, this “root” is actually an artifact of the tree structure, and the placement is an artifact of the NEWICK string representation. In unrooted trees, then the difference in bipartitions as given by the leafset bitmasks is, thus, wholly artifactual. This means that it would be impossible to robustly and reliably compare, relate, and perform any operations on bipartitions coded using leafset bitmasks on unrooted trees: what is effectively the equivalent bipartition of taxa maybe represented either by placing, the first taxon and all the other taxa in the same group as it in group “0” in one representation, or group “1” in another one, and which representation is used is arbitrary and random and unpredictable.

Thus, to allow for robustly establishing equivalence of bipartitions across different representations and instantiations of different unrooted trees, we normalize the bit array representation of bipartitions in unrooted trees to always ensure that the first taxon is assigned to group “0”, whether or not this taxon is actually a descendent or a member of the leafset of the edge. [We also collapse the basal bifurcation of unrooted trees to avoid redundant representation of artifactual bipartitions.]

As the first taxon corresponds to the least-significant bit in the DendroPy scheme, this normalization is known as the least-significant bit 0 or “LSB-0” normalization scheme. The choice of 0 as opposed to 1 is arbitrary, but the reason is so ensure that we can have consistent comparisons of groups across trees of different rotations (and “pseudo-rootings” created by the constraints of tree representation in, e.g., the NEWICK format) by enforcing the convention that group “0” will always be the group that includes the first taxon (i.e., the taxon with index 1, corresponding to the position of the least-significant or right-most bit).

We refer to this normalized version of the leafset bitmask as a split bitmask. For consistency, bipartitions of rooted trees are also assigned split bitmasks, but here these are simply the unmodifed leafset bitmasks. For both unrooted and rooted trees we maintain the leafset bitmask representation in parallel for each bipartition, as this has useful information is lost when normalized, e.g., establishing whether or not a particular subtree or taxon can be found within bipartition.

Thus, regardless of whether the tree is rooted or unrooted, each bipartition on is modeled by two bitmasks: a split bitmask and leafset bitmask. For rooted trees, these are identical in value. For unrooted trees, the split bitmask is the leafset bitmask normalized to constrain the least significant bit to be 0.

A split bitmask is used to establish identity across different trees (for this reason it is also sometimes called a split or bipartition hash), while a leafset bitmask is used to work with various ancestor-descendent relationships within the same tree (it can be used to, for example, quickly assess if a taxon descends from a particular node, or if a particular node is a common ancestor of two taxa).

Leafset bitmasks are unstable representations of bipartitions for unrooted trees, but remain accurate and convenient representations of the descendent leaf-sets of nodes in both unrooted and rooted trees. Split bitmasks, on the other hand, are stable representations of bipartitions for both unrooted as well as rooted trees, but are not accurate representations of the taxa associated with the leaves descended from the bipartition of a particular edge.

## Using Bipartitions¶

### Bipartition Encoding¶

The bipartition encoding of a tree is a specification of the structure of tree in terms of the complete set of bipartitions that can be found on it. Given a bipartition encoding of a tree, the entire topology can be reconstructed completely and accurately. In addition, the bipartition encoding of trees can be used to quickly and accurately compare, relate, and calculate various statistics between different trees and within the same tree.

In DendroPy, the Tree.encode_bipartitions method calculates the bipartitions of a tree. The Edge.bipartition attribute of each edge will be populated by a Bipartition instance, each of which has the bipartition’s split bitmask stored in the Bipartition.split_bitmask attribute and the leafset bitmask stored in the Bipartition.leaf_bitmask attribute. In addition, each Bipartition also stores a reference to the edge to which it corresponds in its Bipartition.edge attribute. For convenience, the split bitmask and the leafset bitmask associated with each bipartition of an edge can be also be accesed through the Edge.split_bitmask and Ede.leafset_bitmask properties, respectively.

You can access these Bipartition objects by iterating over the edges of the tree, but it might be more convenient to access them through the Tree.bipartition_encoding attribute of the Tree. You can also access a dictionary mapping Bipartition instances to their corresponding edges through the Tree.bipartition_edge_map attribute, or a dictionary mapping split bitmasks to their corresponding edges through the Tree.split_bitmask_edge_map attribute.

By default, the Bipartition instances created are immutable. This is to allow them to be used in sets or dictionary keys, and thus exploit O(1) look-up/access performance. The hash value of a Bipartition object is its Bipartition.split_bitmask attribute; two distinct Bipartition objects are considered equivalent even if they refer to different Edge objects on different Tree objects if their Bipartition.split_bitmask values are the same. If you need to modify the values of a Bipartition, you need to set the Bipartition.is_mutable attribute to True. Note that changing any values that modify the hash of a Bipartition instance that is already in a hash container such as a set or dictionary will make that instance or possibly other members of the container inaccessible: never change the value of a Bipartition instance if it is in a set or dictionary.

### Calculating Bipartitions on Trees¶

A large number of DendroPy functions calculate the split and leafset bitmasks in the background: from tree comparison approaches (e.g., calculating the Robinson-Foulds distance), to working with within-tree operations (e.g., finding the most-recent common ancestor between two nodes or patrisitic distances between taxa), to tree-set operations (e.g., building consensus trees or scoring tree leafset credibilities and finding the maximum leafset credibility tree).

When passing trees to these methods and functions, these functions will call Tree.encode_bipartitions automatically for you unless you explicitly specify that this should not be done by passing in ‘is_bipartitions_updated=True‘.

The typical usage idiom in this context would be to:

1. Establish a common taxon namespace [i.e., creating a global TaxonNamespace object and pass it in to all reading/parsing/input operations]
2. Read/load the trees, calling Tree.encode_bipartitions on each one.
3. Perform the calculations, making sure to specify is_bipartitions_updated=True.

For, example, the following snippet shows how you might count the number of trees in a bootstrap file that have the same topology as a tree of interest:

import dendropy
from dendropy.calculate import treecompare
taxa = dendropy.TaxonNamespace()
target_tree = dendropy.Tree.get_from_path(
"mle.tre",
"nexus",
taxon_namespace=taxa)
count = 0
for sup_tree in dendropy.Tree.yield_from_files(
files=["boots1.tre", "boots2.tre", "boostraps3.tre"],
schema="nexus",
taxon_namespace=taxa):
d = treecompare.symmetric_difference(target_tree, sup_tree)
if d == 0:
count += 1
print(count)


For this application, it is simpler just to let the calculations take place in the background. But, for example, if for some reason you wanted to do something more complicated, as it calculating the counts with respect to multiple trees of interest, you should try and avoid the redundant recalculation of the bitmasks:

import dendropy

from dendropy.calculate import treecompare
taxa = dendropy.TaxonNamespace()
tree1 = dendropy.Tree.get_from_path(
"mle1.tre",
"nexus",
taxon_namespace=taxa)
tree1.encode_bipartitions()
tree2 = dendropy.Tree.get_from_path(
"mle2.tre",
"nexus",
taxon_namespace=taxa)
tree2.encode_bipartitions()
counts1 = 0
counts2
for sup_tree in dendropy.Tree.yield_from_files(
files=["boots1.tre", "boots2.tre", "boostraps3.tre"],
schema="nexus",
taxon_namespace=taxa):
sup_tree.encode_bipartitions()
if treecompare.symmetric_difference(
tree1, sup_tree, is_bipartitions_updated=True):
count1 += 1
if treecompare.symmetric_difference(
tree2, sup_tree, is_bipartitions_updated=True):
count2 += 2
print(count1, count2)


Note that in all cases, for bipartitions to be meaningfully compared two conditions must hold:

1. The trees must reference the same operational taxonomic unit namespace object, TaxonNamespace.
2. The trees must have the same rooting state (i.e., all rooted or all unrooted).