Phylogenetic Trees and Multiple Alignments
A tree is an undirected acyclic connected graph. A leaf is an exterior node with degree one. Interior nodes have a degree greater than one.
A phylogenetic tree is a tree, where the leaves are labelled 1..n (with objects, OTUs, species, sequences). One speaks of a tree over n objects. Usually the issue is a binary tree, i.e. every inner node has degree three. Sometimes there are weights on the edges which then represent a distance. A particular edge or node can be highlighted to be the root, the tree is rooted (hierarchy, dendrogram).
Path metric of a weighted tree: if there are positive weights on the edges of a tree over n objects given, the path metric is the metric that results from adding up the weights along a path between to leaves.
When one wants to stress that a tree has no weights attached to the edges one frequently speaks of the topology of the tree.
Regarding a weighted phylogenetic tree, whereat the weights of the edges represent an estimate of the evolutionary distance between the nodes, relies on Darwin's theory of Evolution:
Starting with a set of n known present-day objects a phylogenetic tree may be constructed by first assigning each object a leaf of the tree and then assigning ancestral and unknown objects to the interior nodes.
In the 1960ies Zuckerkandl and Pauling assigned protein sequences to the leaves of a phylogenetic tree being a trace of molecular evolution:
Evolutionary time is reflected in the distances between the
leaves or sequences respectively.
The time has passed during evolution of different species or sequences
starting from a common ancestor.
Thus, assuming that the studied present-day sequences have evolved
from a common ancestor at equal rate, the corresponding phylogenetic tree
would be ultrametric, which implies the existance of a root,
and the same length of all paths from the root to the leaves.
Comments are very welcome.