In a graph $G$, a block is a maximal subgraph which has no cut-vertex or articulation vertex. For example in the given figure there are four blocks; these are induced subgraphs on vertex sets $B_1=\{v_1,v_2,v_3\},\ B_2=\{v_3,v_4\},\ B_3=\{v_3,v_6,v_5, v_7\}, \ B_4=\{v_7,v_8,v_9\}$.

Let graph $G$ has $n$ vertices, and there are $k$ number of blocks $B_1, B_2, \dots, B_k$ having $n_1,n_2,\dots n_k$ number of vertices, respectively. Then, the following relation holds $$n=\sum_{i=1}^{k}(n_i-1)+1.$$ Now, I have following two questions.

Let the blocks $B_1, B_2, \dots, B_k$ have $t_1, t_2, \dots, t_k$ number of cut-vertices of $G$, respectively. Then what can be a relation between $n, ~ n_i$, and $t_i$.

For which combinations of $n_i$, and $t_i$ the following inequality is true. $$\sum_{i=1}^{k}2^{t_i}(n_i-t_i)^{2.81}< n^{2.81}.$$