Consider a plane algebraic curve \(X\) with a singular point \(\xi\) of multiplicity \(m\). A suitable blow-up transforms \(X\) into a curve with at most one point \(\xi _1\) of multiplicity \(m\) in the strict transform of \(X\) which is mapped onto \(\xi\). The point \(\xi _1\) is called an infinitely near singular point. If the procedure is repeated with \(\xi _1\) in the place of \(\xi\), then one may find another infinitely near singular point \(\xi _2\) etc. The sequence of infinitely near singular points \(\xi _1\), \(\xi _2\), \(\ldots\) is always finite, i.e. for some \(\delta \in {\mathbb N}\) there are only points of multiplicity \(<m\) in the strict transform which are mapped onto \(\xi _{\delta}\). The present paper continues the development of differentiation techniques which was started by H. Hironaka [in: Algebraic Geometry, Johns Hopkins Univ. Press, Baltimore, Md. 1976, 52–125 (1977; Zbl 0496.14011)] in 1976 and continued in several further papers. The author proves a finite presentation theorem where the presentation is in terms of a finitely generated graded algebra which describes the total aggregate of the trees of infinitely near singular points.