Is this similarity between trees and vector space bases just a coincidence?

linear-algebra graph-theory

The notion you are looking for are matroids.

Related

Is $\mathbb{S}^\infty$ exotic
Doubling Money Game
How does quater-imaginary (and other imaginary/complex bases) work?
2-Torsion Group Scheme
Is computing distance a lesser capability than computing square roots?
Dimensionality of null space when Trace is Zero
Relation between uniform continuity and uniform convergence
Showing that $ \int_{0}^{\pi/2}\frac{1}{\sqrt{\sin{x}}}\;{dx}=\int_{0}^{\pi/2}\frac{2}{\sqrt{2-\sin^2{x}}}\;{dx}?$
Intuitive understanding of integral of vector valued functions
How far is it true that statements dependent on Axiom of Choice are not constructive.
Is every vector space basis for $\mathbb{R}$ over the field $\mathbb{Q}$ a nonmeasurable set?
Benchmarking programs in Rust