The existence of an operator that maps rational number $1/2$ into the array of Farey tree is proven. It is shown that this operator can be represented by combinatorial compositions of two simple real functions: $f:[0,\, 1]\rightarrow [1/2, \,1]$, which is $(0,\, 1)$-rational and $\sigma:[0,\, 1] \rightarrow [0,\, 1]$, which is linear. Then, another operator, mapping rational $r\in(0,\, 1)$ into the branch of the Farey tree emanating from the node characterized by $r$ is described.