Let $X$ be a a "topological tree" by your definition. Then $X$ is uniquely arcwise connected and Hausdorff. Let $f:S^n\to X$ be a map from the $n$-sphere where $n\geq 1$. It follows from the Hahn-Mazurkiewicz Theorem that the image $f(S^n)$ is a uniquely arc-wise connected Peano continuum. This is equivalent to being a dendrite and dendrites are contractible. Thus $f$ contracts in $f(S^n)$ and it follows that $\pi_n(X)$ is trivial for all $n\geq 0$.

Let $X$ be a a "topological tree" by your definition. Then $X$ is uniquely arcwise connected and Hausdorff. Let $f:S^n\to X$ be a map from the $n$-sphere where $n\geq 1$. It follows from the Hahn-Mazurkiewicz Theorem that the image $f(S^n)$ is a uniquely arcwise connected Peano continuum. This is equivalent to being a dendrite and dendrites are contractible. Thus $f$ contracts in $f(S^n)$ and it follows that $\pi_n(X)$ is trivial for all $n\geq 0$.

The results I'm using here are part of "Continuum Theory." Nadler's book has a great chapter on dendrites.