PHP B-tree - Remove element/node

Here follows an example of deleting from "Multiway B-Tree".

PHP code:

function btree_delete($value, $a, &$higher)
{
 /*  search and delete key $value in B-tree a; if a page underflow arises,
  balance with adjacent page or merge; $higher := "page $a is undersize" */
 
 global $tree, $n;
 
 if( $a === null )
 {
  $higher = false;
 }
 else
 {
  $l = 1;
  $r = $tree[$a]['m'];
  
  // binary search
  do
  {
   $k = (int)( ($l + $r)/2 );
   if( $value <= $tree[$a]['el'][$k]['value'] )
    $r = $k -1;
   if( $value >= $tree[$a]['el'][$k]['value'] )
    $l = $k +1;
  }while( $r >= $l );
  
  
  if( $r === 0 )
  {
   $q = $tree[$a]['p0']; // go to the left
  }
  else
   $q = $tree[$a]['el'][$r]['p']; // go to the right
   
  if( $l - $r > 1)
  {
   // $a is leaf page
   if( $q === null )
   {
    $tree[$a]['m']--;
    $higher = ($tree[$a]['m'] < $n )? true: false;
    
    for( $i=$k; $i <= $tree[$a]['m']; $i++ )
    {
     $tree[$a]['el'][$i] = $tree[$a]['el'][$i+1];
    }
   }
   else
   {
    btree_del($q,$a, $higher, $k);
    if( $higher )
    {
     underflow( $a , $q, $r, $higher, $k); 
    }
   }
  }
  else
  {
   btree_delete( $value, $q, $higher );
   if( $higher )
   {
    underflow( $a, $q, $r, $higher, $k );
   }
  }
 }
 
}

function underflow( $c, $a, $s, &$higher, &$k )
{
 /* $a = underflowing page, $c = ancestor page,
  $s = index of deleted entry in c */
 
 global $tree, $n;
 
 $mc = $tree[$c]['m'];
 
 if( $s < $mc ) 
 {
  $s++;
  
  // $b = page to the right of a
  $b = $tree[$c]['el'][$s]['p'];
  $mb = $tree[$b]['m'];
  
  // $k = nof items available on page b
  $k = (int)( ($mb - $n+1)/2 );
  
  $tree[$a]['el'][$n] = $tree[$c]['el'][$s];
  
  $tree[$a]['el'][$n]['p'] = $tree[$b]['p0'];

  if( $k > 0 )// balance by moving k-1 items from $b to $a
  {
   for( $i=1; $i <= $k-1; $i++ )
   {
    $tree[$a]['el'][$i+$n] = $tree[$b]['el'][$i];
   }
   $tree[$c]['el'][$s] = $tree[$b]['el'][$k];
   $tree[$c]['el'][$s]['p'] = $b;
   
   $tree[$b]['p0'] = $tree[$b]['el'][$k]['p'];
   $mb -= $k;/////// +1
   
   for( $i=1; $i <= $mb; $i++  )
   {
    $tree[$b]['el'][$i] = $tree[$b]['el'][$i+$k];
   }
   $tree[$b]['m'] = $mb;
   $tree[$a]['m'] = $n-1 +$k;
   $higher = false;

  }
  else// merging a and b, discard b
  {
   for( $i=1; $i <= $n; $i++ )
   {
    $tree[$a]['el'][$i+$n] = $tree[$b]['el'][$i];
   }
   for( $i=$s; $i <= $mc-1; $i++)
   {
    $tree[$c]['el'][$i] = $tree[$c]['el'][$i+1];
   }
   $tree[$a]['m'] = 2*$n;
   $tree[$c]['m'] = $mc - 1;
  }
 }
 else
 {
  // $b = page to the left of $a
  if( $s == 1 )
  {
   $b = $tree[$c]['p0'];
  }
  else
  {
   $b = $tree[$c]['el'][$s-1]['p'];
  }
  $mb = $tree[$b]['m']+1;
  $k = (int)( ($mb - $n)/2 );
  
  if( $k > 0 )
  {
   for( $i=$n-1; $i >= 1; $i-- )
   {
    $tree[$a]['el'][$i+$k] = $tree[$a]['el'][$i]; 
   }
   $tree[$a]['el'][$k] = $tree[$c]['el'][$s];
   $tree[$a]['el'][$k]['p'] = $tree[$a]['p0'];

   // move k-1 items from $b to $a, one to $c

   $mb -= $k;//////// +1
   
   for( $i=$k-1; $i >= 1; $i-- )
   {
    $tree[$a]['el'][$i] = $tree[$b]['el'][$i+$mb]; 
   }
   $tree[$a]['p0'] = $tree[$b]['el'][$mb]['p'];
   
   $tree[$c]['el'][$s] = $tree[$b]['el'][$mb];
   $tree[$c]['el'][$s]['p'] = $a;
   
   $tree[$b]['m'] = $mb -1;
   $tree[$a]['m'] = $n-1 +$k;

   $higher = false;
   
  }
  else //merge pages $a and $b, discard $a
  {
   $tree[$b]['el'][$mb] = $tree[$c]['el'][$s];
   $tree[$b]['el'][$mb]['p'] = $tree[$a]['p0'];
   
   for( $i=1; $i <= $n-1; $i++ )
   {
    $tree[$b]['el'][$i+$mb] = $tree[$a]['el'][$i];
   }
   $tree[$b]['m'] = 2*$n;
   $tree[$c]['m'] = $mc - 1;
  }
 }
}

function btree_del($p, $a, &$higher, &$k)
{
 global $tree, $n;
 
 $m = &$tree[$p]['m'];
 
 $q = $tree[$p]['el'][$m]['p'];
 if( $q !== null )
 {
  btree_del($q, $a, $higher, $k);
  if( $higher )
   underflow($p, $q, $m, $higher, $k);
 }
 else
 {
  $tree[$p]['el'][$m]['p'] = $tree[$a]['el'][$k]['p'];
  $tree[$a]['el'][$k] = $tree[$p]['el'][$m];
  
  $m--;

  $higher = ($m < $n )? true: false;
 }
}

Full example of creating and delete from B-Tree here