The author deals with an investigation into the minimum value of \(D= D(n)\) such that any \(n\)-point tree metric space \((T,\rho)\) can be \(D\)-embedded into a given Banach space \((X,\|\cdot\|)\); i.e., there exists a mapping \(f: T\to X\) such that \(D^{-1}\rho(x,y)\leq\|f(x)- f(y)\|\leq \rho(x,y)\) for all \(x,y\in T\). Bourgain showed that \(X\) is super reflexive if and only if \(D(n)\to \infty\) as \(n\to\infty\). For \(X= \ell^p\), \(1< p<\infty\) he gave a lower bound \(\log D(n)\geq\text{const}+ \min\left({1\over 2},{1\over p}\right)\cdot\log(\log\log n)\). The present author presents among other things a more elementary proof of this lower bound and shows that it's tight (up to a multiplicative constant). For Euclidean spaces with no restriction on the dimension \(D(n)= O(\sqrt{\log\log n})\) holds.