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The traversal tree generated by the recursive function without pruning is a binary tree, in which each internal node has exactly two children.
After obtaining NewEst, the QueryInsertion function operated internally in the reader generates a complete binary tree in which the number of leaf nodes equals ⌈NewEst⌉ and returns all leaf nodes into Q (line 11, Figure 1b).
Figure 1 shows a simple example of a complete binary tree in which the CSP conditions are satisfied for every internal node (also see Supplementary Appendix D.6 for a more general example).
We end this section by noting that, in general, the output from the exact linkage algorithm may be a rooted binary tree in which some internal nodes have height zero, - represented co-assignment probabilities of zero.
Suppose we are provided with a set of aligned sequences from M species, whose evolutionary relationship is described by a phylogenetic tree 𝒯=, specified by a collection of nodes 𝒩 and edges ℰ. Assume 𝒯 is a rooted binary tree, in which case the tree contains N=2 M−1 nodes with M−1 being internal (corresponding to ancestral species) and not directly observable.
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(1) DMHBGKA tree is a special binary search tree in which the subtrees of each node differ in height by at most one.
From the Formula (1), which represents the binary tree in a mathematical form, and Figure 3, it can be noticed that the starting position of every fraction is in the middle of two successive slot schedules of the previous fraction.
With this information in mind, it is easy to construct a binary tree of dits and dahs as a mnemonic; Figure 2, which is borrowed from http://www.learnmorsecode.com, shows the binary tree in all its glorious details.
The algorithm takes a non-binary (which is often the case) species tree and during the run reduces it to a binary tree in a variety (or even all) possible ways.
Our findings show that the Barnes–Hut tree outperforms the binary tree in both cases.
The algorithm search works on the concept of the searching a binary tree, which in usual cases yields a performance of ({Theta }(log n+1)).
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