Mutually graceful trees

Mutually graceful trees --- взаимно грациозные деревья.

Let [math]\displaystyle{ T_{p} }[/math] and [math]\displaystyle{ \theta_{p} }[/math] be two trees with vertices [math]\displaystyle{ t_{i} }[/math] and [math]\displaystyle{ u_{i} }[/math] ([math]\displaystyle{ i = 1, 2, \ldots, p }[/math]), respectively; then a labeling [math]\displaystyle{ f }[/math] will be called mutually graceful if it satisfies the following conditions: [math]\displaystyle{ \{f(t_{i})\} \cup \{f(u_{i})\} = \{1, 2, \ldots, 2q\} \mbox{ for } i = 1, 2, \ldots, q(=p-1); }[/math] [math]\displaystyle{ f(t_{p}) = 2q + 1, f(u_{p}) = 2q + 2; }[/math] and the vertex labels of each of the two trees --- with exception of the highest ones defined by (2) --- are at the same time the induced edge labels of the other tree.

Here the induced edge labels are defined as usual:

[math]\displaystyle{ |f(x) - f(y)|\mbox{ for the edge }(x,y). }[/math]