A tree is a connected graph containing no cycles. This simple definition forces the tree to have a branching shape, hence its name. For that matter, a large number of properties "fall out" from defining a tree in this manner, one of which is that for any two nodes of a tree, there exists a unique path connecting the two nodes.
Trees are applied widely, inside and outside of biology. Within biology, perhaps the most common reason for using trees for modeling occurs when constructing phylogenies, which represent the evolutionary connections between a collection of organisms. In phylogenetic analysis, we are most interested in binary trees, for which every node has degree at most 3; this requirement models the Darwinian model of evolution in which species are only created by splitting off from an existing species (usually as the result of an extended period of isolation).