*Forma,* Vol. 8 (No. 2), pp. 277-296, 1993

*Original Paper*

## Algebraic properties of the most parsimonious reconstructions of the hypothetical ancestors on a given tree

Nobuhiro Minaka

Division of Information Analysis, National Institute of Agro-Environmental Sciences, Kannon-dai 3-1-1, Tsukuba, Ibaraki 305, Japan

(Received August 14, 1993; Accepted August 20, 1993)

**Keywords: **
Character evolution, HTU reconstruction, parsimony, partial order, phylogenetics

**Abstract. **
Phylogenetic systematics (or cladistics) is a method for estimating the evolutionary history of organisms based on morphological and molecular characters. For a given set of character information of observed organisms, the most parsimonious cladogram (or phylogenetic tree) which minimizes the total number of character state changes gives the basis upon which any historical analysis of character evolution is to be based. In this paper, I will discuss the implications of the most parsimonious reconstructions of the postulated common ancestors on an unrooted tree. The "most parsimonious reconstruction" (MPR) is the reconstruction (an array of the hypothetical character states) which minimizes the total number of changes on an unrooted tree. There may exist two or more MPRs on a tree for a given set of organisms. For a rooting of an unrooted tree, the set of these equally most parsimonious reconstructions ("MPR-space") can be partially ordered by a binary ordering relation ("MPR partial order"). Several algebraic properties of the solution space ("MPR-poset") are derived from this partial-order formulation. I will conjecture that there exists an MPR ("ACCTRAN": Accelerated Transformation Optimization *sensu* SWOFFORD and MADDISON (1987) which minimizes the length of any subtree. I will propose a new index, "distortion index", which is defined to be the cumulative difference between the lengths of all subtrees of ACCTRAN and those of other MPRs. I will conjecture that another MPR ("DELTRAN": Delayed Transformation Optimization *sensu* SWOFFORD and MADDISON (1987) maximizes the distortion index.