Appendix S1. Branch orders and binary trees
A three-dimensional branched structure can be simplified to a two-dimensional, rooted binary
“tree”, a mathematical object containing nodes and links (i.e., stem segments in the context of
a plant). As binary trees are true bifurcating structures, every node will have only one parent
node and a branching point node may give rise to a maximum of two nodes, except the “root”
(i.e., basal) node, which may only give rise to a single node. It should be noted that here we
are referring, in mathematical terms, to weakly binary trees, where there is no requirement to
consider each side of a branching point as unique, and that binary trees are strictly free of
loops (MacDonald 1983).
There exists an altitude to a rooted binary tree, which is the length of the longest path
(i.e., number of stem segments) between the root node and the terminal nodes (e.g., the
altitude of the binary trees represented in Fig. 1 is 7). A consideration of the positions of
given components of branching within a whole rooted binary tree defines the ordering and
adjacency of nodes (MacDonald 1983). Numerical orders can be assigned to the stem
segments, according to how they connect, working either towards (centripetal or basipetal) or
away from (centrifugal or acropetal) the base of the plant (Berntson 1997). A common
method to define order within a branching structure is to consider the primary stem to be
zero, and modules arising from the primary stem as one, and so on (Fig. 1a). However, this
ordering system can be insensitive to developmental processes, i.e., the top of the primary
stem can be a similar age and function to a module lower in the structure, but has a very
different order. To account for development, we instead used an ordering system that
incremented with each branch event (i.e., is acropetal; Figs. 1b-d); the order of each stem
segment then is simply the topological distance (i.e., number of stem segments) from the base
of the plant, where the root stem segment has the order 1.
The two extreme models of branching in binary trees are known as ‘incomplete’ and
‘perfect’. A perfect binary tree has a true bifurcation at each branch point and each node,
except the root and terminal nodes, has precisely two child nodes. In a perfect binary tree
there are a maximum or near maximum number of stem segments at the depth of the altitude
of the whole tree. In an incomplete binary tree, some nodes at shallower points than the
overall depth of the tree do not have child nodes. Although in the theory of binary trees,
some nodes may have only one child node, in this study only nodes where a branching event
occurred were digitised, so there will not be any one-child nodes, by definition.
R.
raphanistrum exhibits terminal flowering, or racemes, where fruits are borne only on the
terminal stem segments.
No plant is going to approach being a perfect binary tree,
particularly in the later stages of development; if all potential modules developed fully,
botanical branching would be so dense that little light would penetrate into the crown
structure (Aarssen 1995). However, assessment of the relative incompleteness of a plant is a
way of quantifying apical dominance and the resultant effect on reproductive output via
development of terminal racemes.
To measure the relative incompleteness of plants, at the whole plant level, we used
two topological indices: the mean stem segment order (MO), a measure of where in the
branching structure most of the stem has developed, according to the direction of the ordering
system, and the tree asymmetry index (TAI), which provides a measure of where a branching
structure sits in the spectrum between perfect and incomplete binary trees (van Pelt et al.
1989, 1992). These topological indices were first used in a botanical context by Day &
Gould (1997).
Tree asymmetry index (TAI) estimates the degree of asymmetry across an entire
binary tree as the mean of the asymmetry observed at each bifurcation (van Pelt et al., 1992).
At a bifurcation, the degree of asymmetry is an indicator of the dominance of one of the
subsequent sub-trees over the other; at the whole plant level, the mean asymmetry, or TAI, is
a measure of a plant’s tendency towards apical dominance. Furthermore, TAI is independent
of the number of branch components within the binary tree and is thus a powerful tool for
comparing binary trees of varied sizes (van Pelt et al., 1992). At a bifurcation, we consider
b1 and b2 elements in the sub-trees on either side of the bifurcation, and a sum of b distal links
(i.e., b = b1 + b2), indicating the total number of links on both sides of the bifurcation. The
degree to which b1 and b2 differ is a measure of the partition asymmetry, Ap. As this
information is averaged across the entire binary tree, where information from each
bifurcation is weighted equally, the partition asymmetry measure needs to be normalised with
a domain of [0, 1]. Where b1 represents the smaller number of distal links and b2 the larger,
partition asymmetry, Ap, is expressed as
A p (b1 , b2) =
b - 2b1
b-2
where b1>0. Note that Ap > 1 if b1 = 0, but as no single-child nodes were digitised in this
study, this state is not relevant here. When b1 and b2 are equal, a symmetrical bifurcation
results and Ap = 0. When b1 = 1 and b2>1, the bifurcation is highly asymmetrical, with Ap = 1.
TAI is the mean Ap across a binary tree:
1 n1
TAI 
 Ap (b1 , b2)
(q  1) p 1
where q is the total number of terminal links, with q-1 bifurcation points. TAI will also have
the domain [0,1], where TAI near 1 indicates a thin binary tree and values near 0 indicate a
more compact binary tree. TAI for a perfect and two incomplete binary trees, which are
represented in Fig. 1b-1d, are 0, 0.31 and 0.57, respectively, increasing as the trees branch
less and become less complex.
The mean order (MO) is defined as
m
MO 

1
S ( )  
n  0
where n is the total number of links, m is the highest order represented, S(γ) is the number of
links of order γ (van Pelt et al., 1989). The mean order is an overall measure of the amount
of branching within a canopy: in general, a higher mean order will occur when more
branching events have occurred throughout the canopy; a lower mean order will occur with
less branching. The mean order for the perfect and two incomplete binary trees, represented
in Fig. 1b-1d, is 6.06, 4.64 and 4.06, respectively, decreasing as the numbers of links at
higher orders disappear from the binary trees. The mean order is dependent on the tree size
and comparisons between trees of varying sizes can only be made in a qualitative sense. We
also produced kernel density estimates for the number of links in each order.
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Day, J.S. & Gould, K.S. (1997) Vegetative architecture of Elaeocarpus hookerianus.
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van Pelt, J., Verwer, R.W.H. & Uylings, H.B.M. (1989) Centrifugal order distributions in
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van Pelt, J., Uylings, H.B.M., Verwer, R.W.H., Pentney, R.J. & Woldenberg, M.J. (1992)
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Bulletin of Mathematical Biology, 54, 759-784.
Fig. 1. Illustrations of the topological branching systems used to analyse plant architectures
in this study. (a) More common ordering system used by botanists, where there is a
main stem (order 0), primary branches (order 1), and so on; (b-d) example binary trees
showing stem segment incrementing ordering system. (b) is a “perfect” binary tree,
while (c) and (d) are “incomplete” binary trees of contrasting structures. (a) and (c)
are identical trees, but with different branch order notations (see text for details).