Binomial trees are graceful

Figures & data

Fig. 1 Binomial Trees of order 0, 1, 2, 3 and 4.

Table 1 The First and Second half Sequences of $B_{k-1}^{\left(i\right)}$, for $i=1,2$ after the relabeling.

$S_1^{\left(1\right)}:2^{k-1},2^{k-1}+1,\dots ,2^{k-1}+2^{k-2}-1$	$S_1^{\left(2\right)}:0,1,2,\dots ,2^{k-2}-2,2^{k-2}-1$
First half sequence of $B_{k-1}^{\left(1\right)}$	First half sequence of $B_{k-1}^{\left(2\right)}$
$S_2^{\left(1\right)}:2^{k-2},2^{k-2}+1,\dots ,2^{k-1}-1$	$S_2^{\left(2\right)}:2^{k-1}+2^{k-2},\dots ,2^k-2,2^k-1$
Second half sequence of $B_{k-1}^{\left(1\right)}$	Second half sequence of $B_{k-1}^{\left(2\right)}$

Fig. 2 Gracefully labeled Binomial Trees $B_1$ and $B_2$.

Fig. 3 Gracefully labeled Binomial Trees of order 0, 1, 2, 3 and 4.

Fig. 4 Gracefully labeled Binomial Tree of order 5.