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  2. Kruskal's tree theorem - Wikipedia

    en.wikipedia.org/wiki/Kruskal's_tree_theorem

    The version given here is that proven by Nash-Williams; Kruskal's formulation is somewhat stronger. All trees we consider are finite. Given a tree T with a root, and given vertices v, w, call w a successor of v if the unique path from the root to w contains v, and call w an immediate successor of v if additionally the path from v to w contains no other vertex.

  3. Cayley's formula - Wikipedia

    en.wikipedia.org/wiki/Cayley's_formula

    Each labelled rooted forest can be turned into a labelled tree with one extra vertex, by adding a vertex with label n + 1 and connecting it to all roots of the trees in the forest. There is a close connection with rooted forests and parking functions , since the number of parking functions on n cars is also ( n + 1) n − 1 .

  4. Tree (graph theory) - Wikipedia

    en.wikipedia.org/wiki/Tree_(graph_theory)

    A rooted tree T that is a subgraph of some graph G is a normal tree if the ends of every T-path in G are comparable in this tree-order (Diestel 2005, p. 15). Rooted trees, often with an additional structure such as an ordering of the neighbors at each vertex, are a key data structure in computer science; see tree data structure.

  5. Recursive tree - Wikipedia

    en.wikipedia.org/wiki/Recursive_tree

    The number of size-n recursive trees is given by = ()!. Hence the exponential generating function T(z) of the sequence T n is given by =! = ⁡ ().Combinatorically, a recursive tree can be interpreted as a root followed by an unordered sequence of recursive trees.

  6. Tree (set theory) - Wikipedia

    en.wikipedia.org/wiki/Tree_(set_theory)

    Trees with a single root may be viewed as rooted trees in the sense of graph theory in one of two ways: either as a tree (graph theory) or as a trivially perfect graph. In the first case, the graph is the undirected Hasse diagram of the partially ordered set, and in the second case, the graph is simply the underlying (undirected) graph of the ...

  7. Cartesian tree - Wikipedia

    en.wikipedia.org/wiki/Cartesian_tree

    Cartesian trees are defined using binary trees, which are a form of rooted tree.To construct the Cartesian tree for a given sequence of distinct numbers, set its root to be the minimum number in the sequence, [1] and recursively construct its left and right subtrees from the subsequences before and after this number, respectively.

  8. List of phylogenetic tree visualization software - Wikipedia

    en.wikipedia.org/wiki/List_of_phylogenetic_tree...

    an online tool for phylogenetic tree view (newick format) that allows multiple sequence alignments to be shown together with the trees (fasta format) EvolView [3] an online tool for visualizing, annotating and managing phylogenetic trees IcyTree [4] Client-side Javascript SVG viewer for annotated rooted trees. Also supports phylogenetic networks

  9. m-ary tree - Wikipedia

    en.wikipedia.org/wiki/M-ary_tree

    The left chain of T is a sequence of ,, …, nodes such that is the root and all nodes except have one child connected to their left most (i.e., []) pointer. Any m- ary tree can be transformed to a left-chain tree using sequence of finite left-t rotations for t from 2 to m .

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