Sackin Index in Phylogenetics

• The Sackin index is a classical statistic that measures rooted tree imbalance by summing the depths of all leaves.
• It can be computed in linear time via depth-first or breadth-first search and recurses naturally with tree decompositions.
• Its various formulations and asymptotic behaviors under models like Yule and uniform make it key for evaluating tree shape and evolutionary dynamics.

Searching arXiv for recent and foundational papers on the Sackin index to ground the article in published work. The Sackin index is a classical statistic of rooted tree balance, defined as the sum of the depths of all leaves in a rooted phylogenetic tree. For a rooted tree $T$ with root $\rho$, leaf set $V_L(T)$, and depth function $\delta_T(v)$ equal to the number of edges on the unique path from $\rho$ to $v$, it is given by

$S(T)=\sum_{x\in V_L(T)} \delta_T(x).$

In phylogenetics it is used to quantify imbalance, compare observed trees with stochastic null models, and study extremal tree shapes; in computer science it coincides with total external path length. A substantial literature develops equivalent formulations, exact expectations under Yule and uniform models, extremal characterizations, algorithmic recursions, and several extensions from trees to phylogenetic networks (Fischer et al., 10 Jun 2025, Mir et al., 2012).

1. Definition, conventions, and scope

The standard convention places the root at depth $0$, so each leaf contributes its graph distance from the root. Under this convention the Sackin index is a function only of rooted tree topology: it is invariant under isomorphisms of pending subtrees, under relabeling of leaves, and independent of edge lengths. In the rooted-binary phylogenetic setting, leaves are bijectively labeled by a taxon set, but the value of $S(T)$ depends only on the underlying shape (Fischer et al., 10 Jun 2025).

Several closely related conventions appear in the literature. If depths are counted with the root at depth $1$ rather than $\rho$0, then every leaf depth increases by $\rho$1, so the Sackin index shifts by $\rho$2, where $\rho$3 is the number of leaves: $\rho$4 Likewise, in planted-tree conventions for phylogenetic networks, the planted root contributes an additional edge to every root-to-leaf path, again producing an additive $\rho$5-shift relative to non-planted conventions. These differences are purely conventional, but they matter when comparing formulas across papers (Fischer, 2018, Zhang, 2021).

A normalized form also occurs. The average leaf depth,

$\rho$6

induces the same ordering as $\rho$7 when $\rho$8 is fixed and is the quantity called “depth” in some empirical and model-comparison studies. This normalization is especially useful when comparing trees of different sizes (Fischer et al., 10 Jun 2025, Keller-Schmidt et al., 2011).

2. Equivalent formulations, recursion, and computation

A fundamental equivalence rewrites the Sackin index as a sum over internal vertices. If $\rho$9 or $V_L(T)$0 denotes the number of descendant leaves in the pending subtree rooted at an internal vertex $V_L(T)$1, then

$V_L(T)$2

for arbitrary rooted trees. The equality follows by double-counting ancestor–leaf pairs: each leaf contributes $V_L(T)$3 for every internal ancestor on its root-to-leaf path, and each internal vertex contributes once for each descendant leaf (Fischer et al., 10 Jun 2025, Mir et al., 2012).

This equivalence has several consequences. First, it makes explicit that the Sackin index can be interpreted either as a leaf-depth statistic or as a clade-size statistic. Second, it yields immediate recursions. If an arbitrary rooted tree decomposes at the root as $V_L(T)$4 with $V_L(T)$5, then

$V_L(T)$6

For a binary tree with left and right pending subtrees $V_L(T)$7 and $V_L(T)$8, this specializes to the standard recurrence

$V_L(T)$9

This recursion is one of the central structural identities behind exact calculations and dynamic-programming algorithms (Fischer et al., 10 Jun 2025).

The index is also local. If a pending subtree $\delta_T(v)$0 is replaced by another pending subtree $\delta_T(v)$1 with the same leaf count and the same attachment depth, then

$\delta_T(v)$2

This locality can be read either from the affine leaf-depth formulation or from the clade-size formulation. It is important both conceptually and algorithmically, because it allows updates under local tree edits to be reduced to the modified region (Fischer et al., 10 Jun 2025).

From an algorithmic perspective, two linear-time computations are standard. In the leaf-depth route, a single DFS or BFS from the root accumulates $\delta_T(v)$3 over leaves. In the clade-size route, a postorder traversal computes $\delta_T(v)$4 for each internal vertex and then sums these values. For trees without degree-1 inner vertices, this is $\delta_T(v)$5 in the number of leaves (Fischer et al., 10 Jun 2025, Mir et al., 2012).

3. Extremal values and minimizing or maximizing shapes

On rooted binary trees, the unique maximizer of the Sackin index is the caterpillar tree $\delta_T(v)$6. Its value is

$\delta_T(v)$7

Equivalently, the leaf-depth multiset of the rooted binary caterpillar is $\delta_T(v)$8, and summing those depths yields the same formula. This is the standard “most imbalanced” extremal case (Fischer et al., 10 Jun 2025, Fischer, 2018).

The binary minimum is subtler. Let

$\delta_T(v)$9

Then the minimum Sackin value on binary trees is

$\rho$0

which reduces to $\rho$1 when $\rho$2. The minimizing trees are precisely those binary trees whose leaves occur only at depths $\rho$3 and $\rho$4. In particular, when $\rho$5 is a power of two, the fully balanced tree is the unique minimizer; for general $\rho$6, the minimum is typically attained by multiple shapes, including the maximally balanced tree $\rho$7 and the greedy-from-the-bottom tree $\rho$8 (Fischer et al., 10 Jun 2025, Fischer, 2018).

The non-binary setting changes the minimum but not the maximum. If multifurcations are allowed, the unique minimizer is the star tree $\rho$9, in which every leaf has depth $v$0, so

$v$1

By contrast, even in the non-binary setting the unique maximizer remains the binary caterpillar, because refining multifurcations into a chain can strictly increase leaf depths (Fischer et al., 10 Jun 2025, Fischer, 2018).

A useful structural comparison with the Colless index is also known: every Colless-minimal rooted bifurcating tree is Sackin-minimal, but the converse is false. Thus the Sackin-minimal class is strictly larger than the Colless-minimal class. This is one precise sense in which the Sackin index is coarser as a discriminator of balance extremes (Coronado et al., 2019).

4. Expected values under stochastic tree models

For random phylogenetic trees, the Sackin index is a central model-testing statistic because different generative mechanisms produce different growth orders. Under the Yule, or Equal-Rate Markov, model on rooted binary phylogenetic trees,

$v$2

where $v$3 is the $v$4-th harmonic number. Asymptotically,

$v$5

so the expected imbalance grows on the order $v$6 (Mir et al., 2012, Cardona et al., 2012).

The Yule model also admits an exact variance formula: $v$7 where $v$8. The corresponding exact second moment is

$v$9

These formulas enable exact standardization of $S(T)=\sum_{x\in V_L(T)} \delta_T(x).$0 under the Yule null for every $S(T)=\sum_{x\in V_L(T)} \delta_T(x).$1 (Cardona et al., 2012).

Under the uniform model on labeled rooted binary phylogenetic trees, all trees in $S(T)=\sum_{x\in V_L(T)} \delta_T(x).$2 have probability $S(T)=\sum_{x\in V_L(T)} \delta_T(x).$3. Earlier work established the asymptotic law

$S(T)=\sum_{x\in V_L(T)} \delta_T(x).$4

and an explicit exact formula was later derived: $S(T)=\sum_{x\in V_L(T)} \delta_T(x).$5 The derivation counts trees by the depth of a fixed leaf, using ordered forests and symmetry over leaf labels. This was presented as the first explicit closed-form expression for the expected Sackin index under the uniform model, refining the earlier asymptotic statement (Mir et al., 2012). A later paper also gave a short elementary proof of a closed formula for the same expectation in the labeled uniform setting (Goh et al., 2022).

The distinction between models is qualitative as well as quantitative. The uniform labeled model yields $S(T)=\sum_{x\in V_L(T)} \delta_T(x).$6-growth, whereas the Yule model yields $S(T)=\sum_{x\in V_L(T)} \delta_T(x).$7-growth; this reflects the fact that the uniform model typically produces more imbalanced trees than the Yule model. For tree shapes, meaning rooted full binary unlabeled trees sampled uniformly, singularity analysis gives

$S(T)=\sum_{x\in V_L(T)} \delta_T(x).$8

with $S(T)=\sum_{x\in V_L(T)} \delta_T(x).$9, so the $0$0 order persists in the unlabeled-shape setting as well (Mir et al., 2012, Goh et al., 2022).

The Sackin index has also been used to diagnose nonclassical branching mechanisms. In an innovation-driven branching process, the average leaf depth $0$1 scales as $0$2, so

$0$3

This is markedly deeper than both Yule and uniform-tree behavior and was reported to be compatible with empirical phylogenies in the datasets analyzed there (Keller-Schmidt et al., 2011).

5. Relation to other balance indices

The Sackin and Colless indices are the two most widely used classical rooted-binary imbalance indices. Both are topology-based, both are invariant under isomorphisms and leaf relabelings, and both attain their maximum values at rooted caterpillars. However, they are not equivalent statistics: Colless measures local split asymmetry, whereas Sackin aggregates leaf depths or clade sizes (Mir et al., 2012).

A basic identity links the Sackin index to the total cophenetic index $0$4 and the total area $0$5: $0$6 where

$0$7

This identity holds for all rooted phylogenetic trees and provides a direct bridge between depth-based and LCA-based balance summaries. It was used to derive exact expectations for $0$8 once exact information for $0$9 and $S(T)$0 was available (Mir et al., 2012).

Resolution power is one of the main points of contrast. The total cophenetic index has a larger value range and greater resolution power than Sackin’s or Colless’s indices, and Sackin’s index can be unchanged under certain cousin interchanges that alter perceived symmetry. Concrete examples were given in which distinct non-isomorphic shapes share the same minimal Sackin value, and the same phenomenon can occur for Colless. This limited sensitivity is one reason the Sackin index is often used alongside, rather than instead of, other balance indices (Mir et al., 2012).

Recent structural work formalizes these relationships in two ways. First, the metaconcept framework shows that the Sackin index is simultaneously a first-order leaf-depth metaconcept and a first-order clade-size metaconcept: $S(T)$1 Affine perturbations of these function classes preserve equivalence to $S(T)$2 under the conditions stated in the framework, which places Sackin at the affine boundary between convex and concave tree-shape functionals (Fischer et al., 10 Jun 2025). Second, a decomposition into two elementary internal-node sums,

$S(T)$3

with $S(T)$4, yields

$S(T)$5

In this representation, the difference $S(T)$6 is exactly $S(T)$7. This makes explicit that Sackin and Colless are compound indices assembled from the same two building blocks, but with different signs (Knüver et al., 5 Sep 2025).

6. Extensions to networks and generalized settings

The classical Sackin index is defined for rooted trees, where each leaf has a unique root-to-leaf path. In phylogenetic networks, multiple directed paths may exist, so extensions are not unique. One extension for simplex networks defines the depth of a node $S(T)$8 as the length of the longest directed path from the root to $S(T)$9,

$1$0

and then sets

$1$1

Under the uniform model on simplex networks, the expected network Sackin index satisfies

$1$2

with explicit bounds

$1$3

This growth is strictly larger than the $1$4 behavior of uniformly sampled rooted binary trees (Zhang, 2021).

For galled trees, two canonical extensions are based on maximal and minimal root-to-leaf path lengths. Writing

$1$5

the extended indices are

$1$6

For uniformly sampled labeled and unlabeled galled trees, simplex galled trees, and normal galled trees, both extensions have mean asymptotic form

$1$7

with explicit class-dependent constants, and the scaled index converges in distribution and with all moments to the Airy distribution (Fuchs et al., 2024).

A different extension arises from entropy-based balance indices for networks. The $1$8 family satisfies $1$9, and for binary trees its expansion at $\rho$00 recovers the classical Sackin index from the derivative. This motivates the network definition

$\rho$01

which agrees with the classical Sackin index on binary trees, although it does not match the usual Sackin definition for multifurcating trees. This shows that “the Sackin index of a network” is now a family of related constructions rather than a single universally adopted object (Bienvenu et al., 23 Jun 2026).

Taken together, these developments place the Sackin index in a broad methodological landscape. At its core it remains the sum of leaf depths, equivalently the sum of internal clade sizes, on rooted trees. Around that core lies a mature theory of extremal shapes, exact expectations, locality, linear-time computation, and model discrimination, together with several principled but nonequivalent extensions to phylogenetic networks (Fischer et al., 10 Jun 2025, Fuchs et al., 2024).