Algorithm 1. Pseudo-code for k-tree construction

input: D output: $T$ 1. $T_{\pi }$ ← create a new node with $S_{T_{\pi }}=\varnothing$ 2. for $d\in$D 3.   $T_{\pi }$ ← $F$($T_{\pi },d, 1)$ 4. end for 5. function $F\left(n,d,i\right)$ 6.   $j \leftarrow 1$ 7.   while i ≤ |$d$| do 8.    if $j$ > $\left|S_n\right|$then            //full match, so travel deeper// 9.      if $d\left[i\right]$ = “0” then          //travel left child// 10.      $n_l$ ← $F$ ($n_l$,$d, i$) 11.     else if $d\left[i\right]$ = “1” then     //travel right child// 12.      $n_r$ ← $F$ ($n_r$, d$,i$) 13.     end if 14.     break while 15.    else if $S_n\left[j\right]$ ≠ $d\left[i\right]$ then      //partial match, so do splitting // 16.     $m$ ← a new node            //create a new node for splitting// 17.      if $d\left[i\right]$ = “0” then           //create a left child// 18.      $m_l$ ← create a new node 19.      $m_r$ ← $n$ 20.     else if $d\left[i\right]="$1” then       //create a right child// 21.      $m_r$ ← create a new node 22.      $m_l$ ← $n$ 23.     end if 24.     adjust $S_m$, $S_{m_r}$, $S_{m_l}$ for nodes m, $m_r$, and $m_l$ 25.     $n \leftarrow m$ 26.     break while 27.    end if 28.     $i\leftarrow i+1; j\leftarrow j+1$ 29.   end while 30.   output $n$ 31. end function