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We derive sharp lower bounds for the first and the second Zagreb indices($M_1$ and $M_2$ respectively) for trees and chemical trees with the givennumber of pendent vertices and find optimal trees. $M_1$ is minimized by a treewith all internal vertices having degree 4, while $M_2$ is minimized by a treewhere each "stem" vertex is incident to 3 or 4 pendent vertices and oneinternal vertex, while the rest internal vertices are incident to 3 otherinternal vertices. The technique is shown to generalize to the weighted firstZagreb index, the zeroth order general Randi\'{c} index, as long as to manyother degree-based indices.  Later the erratum was added: Theorem 3 says that the second Zagreb index$M_2$ cannot be less than $11n-27$ for a tree with $n\ge 8$ pendent vertices.Yet the tree exists with $n=8$ vertices (the two-sided broom) violating thisinequality. The reason is that the proof of Theorem 3 relays on a tacitassumption that an index-minimizing tree contains no vertices of degree 2. Thisassumption appears to be invalid in general. In this erratum we show that theinequality $M_2 \ge 11n-27$ still holds for trees with $n\ge 9$ vertices andprovide the valid proof of the (corrected) Theorem 3.

Minimizing Degree-based Topological Indices for Trees with Given Number of Pendent Vertices + Erratum