Exrermal trees with respect to the Sackin index

Exrermal trees with respect to the Sackin index

The Sackin index of a rooted tree T that summarizes the shape of T is defined as the sum of the depths of its leaves and is denoted by S(T). The Sackin index measures the degree of balance of rooted phylogenetic trees and acyclic molecular graphs. For a given number n of leaves (terminal taxa), extermal phylogenetic and binary trees with exactly n leaves and with respect to the Sackin index and Colless index are characteized in the literature. In this paper we show that n − 1 ≤ S(T) ≤ ⌊n/2⌋ ⌈n/2⌉ for each rooted n-vertex tree T and we characteize all rooted n-ertex trees with extermal Sackin index.Also, we provide exact lower and upper bounds for the Sackin index of rooted d–ary trees and we characteize extermal families of them.

The Sackin index of a rooted tree T that summarizes the shape of T is defined as the sum of the depths of its leaves and is denoted by S(T). The Sackin index measures the degree of balance of rooted phylogenetic trees and acyclic molecular graphs. For a given number n of leaves (terminal taxa), extermal phylogenetic and binary trees with exactly n leaves and with respect to the Sackin index and Colless index are characteized in the literature. In this paper we show that n − 1 ≤ S(T) ≤ ⌊n/2⌋ ⌈n/2⌉ for each rooted n-vertex tree T and we characteize all rooted n-ertex trees with extermal Sackin index.Also, we provide exact lower and upper bounds for the Sackin index of rooted d–ary trees and we characteize extermal families of them.