Notes on Natural Logic Notes for PHIL370

A tree is a structure T = (T,E), where T is a nonempty set whose elements are called nodes and E is a relation on T , E ⊆ T × T , called the immediate edge relation, satisfying the following conditions: for all nodes, n, n′,m ∈ T , • Every node has a unique predecessor: If nEm and n′Em, then n = n′, • There are no cycles: If (n1, . . . , nk) is a sequence of nodes where for each i = 1, . . . , k − 1, niEni+1, then n1 6= nk, and • Between any two nodes there is a unique path: For each n, n′ ∈ T there is a unique sequence n1, n2, . . . , nm such that n = n1En2 · · ·Enm = n′. Let n be a node, then succ(n) = {n′ | nEn′} are the successors of n and pred(n) = {n′ | n′En} are the predecessors of n. A node r is called the root of the tree provided pred(r) = ∅. A node n is called a leaf provided succ(n) = ∅. A path in a tree is a sequence of nodes connected by an edge relation: a path is a sequence (n1, n2, . . . , nk) such that for each i = 1, . . . , k − 1, niEni+1. The length of a path is equal to the number of edges along that path (equivalently, one minus the number of nodes). The height of a tree is the length of the longest path. Here is an example: