TreeNumber: Unique integer indices for bifurcating tree topologies

In TreeTools: Create, Modify and Analyse Phylogenetic Trees

Unique integer indices for bifurcating tree topologies

Description

Functions converting between phylogenetic trees and their unique decimal representation, based on a concept by John Tromp, employed in \insertCiteLi1996;textualTreeTools.

Usage

as.TreeNumber(x, ...)

## S3 method for class 'phylo'
as.TreeNumber(x, ...)

## S3 method for class 'multiPhylo'
as.TreeNumber(x, ...)

## S3 method for class 'character'
as.TreeNumber(x, nTip, tipLabels = TipLabels(nTip), ...)

## S3 method for class 'TreeNumber'
as.TreeNumber(x, ...)

## S3 method for class 'MixedBase'
as.TreeNumber(x, ...)

## S3 method for class 'TreeNumber'
as.MixedBase(x, ...)

## S3 method for class 'integer64'
as.MixedBase(x, tipLabels = NULL, ...)

## S3 method for class 'numeric'
as.MixedBase(x, tipLabels = NULL, ...)

## S3 method for class 'numeric'
as.phylo(x, nTip = attr(x, "nTip"), tipLabels = attr(x, "tip.label"), ...)

## S3 method for class 'TreeNumber'
as.phylo(x, nTip = attr(x, "nTip"), tipLabels = attr(x, "tip.label"), ...)

as.MixedBase(x, ...)

## S3 method for class 'MixedBase'
as.MixedBase(x, ...)

## S3 method for class 'phylo'
as.MixedBase(x, ...)

## S3 method for class 'multiPhylo'
as.MixedBase(x, ...)

## S3 method for class 'MixedBase'
as.phylo(x, nTip = attr(x, "nTip"), tipLabels = attr(x, "tip.label"), ...)

Arguments

x	Integer identifying the tree (see details).
...	Additional parameters for consistency with S3 methods (unused).
nTip	Integer specifying number of leaves in the tree.
tipLabels	Character vector listing the labels assigned to each tip in a tree, perhaps obtained using TipLabels().

Details

There are NUnrooted(n) unrooted trees with n leaves. As such, each n-leaf tree can be uniquely identified by a non-negative integer x < NUnrooted(n).

This integer can be converted by a tree by treating it as a mixed-base number, with bases 1, 3, 5, 7, … (2 n - 5).

Each digit of this mixed base number corresponds to a leaf, and determines the location on a growing tree to which that leaf should be added.

We start with a two-leaf tree, and treat 0 as the origin of the tree.

0 ---- 1

We add leaf 2 by breaking an edge and inserting a node (numbered 2 + nTip - 1). In this example, we'll work up to a six-leaf tree; this node will be numbered 2 + 6 - 1 = 7. There is only one edge on which leaf 2 can be added. Let's add node 7 and leaf 2:

0 ---- 7 ---- 1
       |
       |
       2

There are now three edges on which leaf 3 can be added. Our options are:

Option 0: the edge leading to 1;

Option 1: the edge leading to 2;

Option 2: the edge leading to 7.

If we select option 1, we produce:

0 ---- 7 ---- 1
       |
       |
       8 ---- 2
       |
       |
       3

1 is now the final digit of our mixed-base number.

There are five places to add leaf 4:

Option 0: the edge leading to 1;

Option 1: the edge leading to 2;

Option 2: the edge leading to 3;

Option 3: the edge leading to 7;

Option 4: the edge leading to 8.

If we chose option 3, then 3 would be the penultimate digit of our mixed-base number.

If we chose option 0 for the next two additions, we could specify this tree with the mixed-base number 0021. We can convert this into decimal:

0 × (1 × 3 × 5 × 9) +

0 × (1 × 3 × 5) +

3 × (1 × 3) +

1 × (1)

= 10

as.TreeNumber() supports up to 51 leaves. For trees with at most 19 leaves, the number fits in a 64-bit integer and the TreeNumber is stored as an integer64 (via the bit64 package), enabling arithmetic and exact round-tripping via as.MixedBase(). For trees with 20–51 leaves, there are more than 2^64 distinct topologies, so the tree number is stored as a decimal character string instead.

Package developers can use the C++ header TreeTools/tree_number.h (via LinkingTo: TreeTools) for the underlying 256-bit encoding (tree_num_t) directly.

Value

as.TreeNumber() returns an object of class TreeNumber with attributes nTip and tip.label. For trees with at most 19 leaves the underlying storage is a single integer64 value (class c("TreeNumber", "integer64")), enabling integer64 arithmetic and exact round-tripping through as.MixedBase(). For trees with 20–51 leaves the number exceeds 2^64, so it is stored as a decimal character string (class c("TreeNumber", "character")). If x is a list of trees or a multiPhylo object, as.TreeNumber() returns a corresponding list of TreeNumber objects.

as.phylo.numeric() returns a tree of class phylo.

Author(s)

Martin R. Smith ([e-mail])

References

\insertAllCited

See Also

Describe the shape of a tree topology, independent of leaf labels: TreeShape()

Other tree generation functions: ConstrainedNJ(), GenerateTree, NJTree(), TrivialTree

Other 'TreeNumber' utilities: is.TreeNumber(), print.TreeNumber()

Examples

tree <- as.phylo(10, nTip = 6)
plot(tree)
as.TreeNumber(tree)

# Trees with 20--51 leaves are stored as decimal strings:
as.TreeNumber(BalancedTree(19))  # integer64-backed
as.TreeNumber(BalancedTree(51))  # character-backed

# If > 9 digits, represent the tree number as a string.
treeNumber <- as.TreeNumber("1234567890123", nTip = 14)
tree <- as.phylo(treeNumber)
as.phylo(0:2, nTip = 6, tipLabels = letters[1:6])

TreeTools documentation built on April 23, 2026, 5:06 p.m.