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The transmission of a vertex $v$ of a graph $G$ is the sum of distances from$v$ to all the other vertices in $G$. A graph is transmission irregular if allof its vertices have pairwise different transmissions. A starlike tree$T(k_1,\ldots,k_t)$ is a tree obtained by attaching to an isolated vertex $t$pendant paths of lengths $k_1,\ldots,k_t$, respectively. It is proved that if astarlike tree $T(a,a+1,\ldots,a+k)$, $k\ge 2$, is of odd order, then it istransmission irregular. $T(1,2,\ldots,\ell)$, $\ell \ge 3$, is transmissionirregular if and only if $\ell \notin \{r^2 + 1:\ r\ge 2\}$. Additionalinfinite families among the starlike trees and bi-starlike trees aredetermined. Transmission irregular unicyclic infinite families are alsopresented, in particular, the line graph of $T(a,a+1,a+2)$, $a\ge 2$, istransmission irregular if and only if $a$ is even.

Constructing new families of transmission irregular graphs