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. 2022 Apr 12;71(5):1210–1224. doi: 10.1093/sysbio/syac027

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© The Author(s) 2022. Published by Oxford University Press on behalf of the Society of Systematic Biologists.

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (https://creativecommons.org/licenses/by-nc/4.0/), which permits non-commercial re-use, distribution, and reproduction in any medium, provided the original work is properly cited. For commercial re-use, please contact journals.permissions@oup.com

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Figure 3. — a) A tree in which each internal node has null size and splits its descendants into subtrees of equal magnitude, and hence . This tree can be considered balanced only according to an index that accounts for node size. b) A linear tree, for which . c–e) A robust, universal tree balance index is insensitive to the addition of a subtree of arbitrarily small magnitude if it is added to a leaf (a) or a nonroot node with out-degree 1 (b), but not necessarily if the subtree is added to a nonroot node with greater out-degree (c).

Inline graphic — a) A tree in which each internal node has null size and splits its descendants into subtrees of equal magnitude, and hence . This tree can be considered balanced only according to an index that accounts for node size. b) A linear tree, for which . c–e) A robust, universal tree balance index is insensitive to the addition of a subtree of arbitrarily small magnitude if it is added to a leaf (a) or a nonroot node with out-degree 1 (b), but not necessarily if the subtree is added to a nonroot node with greater out-degree (c).