Regarding proper form of production rules of Context-free tree grammars
After a couple of hours of thinking I found at least proper form of grammar. Thanks to @Sylvain's commentary and link to Fisher's paper, which given me a clue.

Proper context-free tree grammar, which yield language coincides with $\{a^nb^nc^n|n&gt;0\} $ will have simplest form:

$G=\{\Sigma,N,S,P\}$, where $\Sigma=\{a,b,c\}$, $N=\{S_t,F,A,B,C\}$, $S=\{S_t\}$, and set of production rules:

$S_t\rightarrow F(a,b,c)\\
F(x_1,x_2,x_3)\rightarrow F(A(a,x_1),B(b,x_2),C(c,x_3))$

That is it. Core idea is to wrap "stack" into subtrees of the form $A(a,x_1)$.

Thereby we receive following productions:


$S_t\Rightarrow F(a,b,c)$, $yield(F(a,b,c))=abc$
$S_t\Rightarrow F(a,b,c)\Rightarrow F(A(a,a),B(b,b),C(c,c))$, $yield(F(A(a,a),B(b,b),C(c,c)))=aabbcc$
$S_t\Rightarrow F(a,b,c)\Rightarrow F(A(a,a),B(b,b),C(c,c))\Rightarrow F(A(a,A(a,a)),B(b,B(b,b)),C(c,C(c,c)))$,
and so on.