Tree-component expectation in the Erdős-Rényi model ($\mathbb{E}{X}_j=\left(\genfrac{}{}{0ex}{}{n}{j}\right)j^{j-2}{p}^{j-1}(1-p{)}^{j(n-j)+\left(\genfrac{}{}{0ex}{}{j}{2}\right)-j+1}$)

For $X_j$ the number of tree components of order $j$ in a binomial random graph, choose their vertices, choose one of $j^{j-2}$ labelled trees by the Cayley formula, require its $j-1$ edges, and exclude both the remaining internal edges and all crossing edges. For $j=1$ interpret $j^{j-2}=1$, recovering the isolated vertex count. If $p=\lambda /n$ for fixed positive $\lambda$ and $j=O(\log n)$, the Stirling formula gives $\mathbb{E}{X}_j\sim n(\lambda e^{1-\lambda }{)}^j/(\lambda \sqrt{2\pi }{j}^{5/2})$. Distinct overlapping vertex sets cannot both be graph components; for disjoint sets their joint occurrence gains the factor $(1-p{)}^{-j^2}$ relative to the product, because their between-set edges must be absent only once.

For fixed $k\ge 2$ and $\omega \to \infty$, the expected number of tree components of order $k$ is asymptotic to $k^{k-2}{n}^k{p}^{k-1}{e}^{-knp}/k!$ throughout the displayed window. It tends to infinity: split $np$ at $1$ and $(\log n)/(2k)$ to obtain lower bounds of orders ${\omega }^{k-1}$, $\sqrt{n}$, and $e^{\omega }$. Distinct overlapping vertex sets cannot both be components; disjoint sets have joint-to-product probability ratio $(1-p{)}^{-k^2}=1+o(1)$. Thus the relative variance vanishes, proving a component of order $k$ exists with high probability.