The book attempts to describe the world of graph theory with emphasis on count-ing of specific kinds of graphs, and in particular trees. A specific focus will be givento its applications in biology, in particular, in phylogeny. It will embed questions inthe context of counting graphs in general.The book only use methods which are present in the first classes of undergraduatestudies. A background in in elementary set theory, mathematical logic, linear algebra,probability theory and calculus is assumed. If several facts about discrete and com-binatorial mathematics are not present for the reader, the book includes an appendixwith the most of the important results in short reviews. In such a restricted sense thepresent script can be read as an introduction into Discrete Mathematics with focusin Graph Theory. Additionally, references for further reading are given.As an textbook the present script contains several exercises, but there are vast dif-ferences in level of these questions: a) Exercises which are straightforward from thetext; b) Problems which need a longer discussion; and c) Open tasks.