  A configuration of the lit-only $\sigma$-game on a finite graph $\Gamma$ is
an assignment of one of two states, on or off, to all vertices of $\Gamma.$
Given a configuration, a move of the lit-only $\sigma$-game on $\Gamma$ allows
the player to choose an on vertex $s$ of $\Gamma$ and change the states of all
neighbors of $s.$ Given any integer $k$, we say that $\Gamma$ is $k$-lit if,
for any configuration, the number of on vertices can be reduced to at most $k$
by a finite sequence of moves. Assume that $\Gamma$ is a tree with a perfect
matching. We show that $\Gamma$ is 1-lit and any tree obtained from $\Gamma$ by
adding a new vertex on an edge of $\Gamma$ is 2-lit.
