Preliminaries and Notation Sample Clauses

Preliminaries and Notation. A binary rooted tree T is full if every node in the tree have either 2 or 0 children. We denote by V (T ) and E(T ) the set of its nodes and edges, respectively, and with L(T ) the leaves set of T . If v ∈ V (T ) and v is not the root of T , then p(v) denotes its father and s(v) its sibling. Given two nodes u, v ∈ V (T ), u is said an ancestor of v, denoted by u ≤T v, if either u ≡ v or v is contained in the subtree rooted at u. If either u ≤T v or v ≤T u, then we call them comparable nodes and write u ∼=T v. We say that u and v are incomparable if they are not comparable and write u ÇT v. The least common ancestor lca(S) of a set of nodes S ⊆ V (T ) is the ≤T -minimal node u s.t. u ≤T s for all s ∈ S. We denote by pathT (u, v) = (t1,..., tj) the (unique) ordered sequence of nodes of T that must be traversed to reach v from u; of course t1 ≡ u and tj ≡ v. The length of pathT (u, v) is j − 1 and it is denote as |pathT (u, v)|. Let Tv be the subtree of T rooted at v and let e = (u, v) be an edge of T , we introduce here the tree Te (see Figure 2.10) as the subtree of T obtained from T : - eliminating all nodes and edges of Tv; - if u is not the root of T , substituting edges (p(u), u) and (u, s(v)) with the single edge (p(u), s(v)), otherwise simply eliminating edge (u, s(v)). When there is ambiguity on the edge e = (u, v) we denote the tree Te as Tu.
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