Understanding CLRS BFT lemma
Suppose you have a tree like

        a           &lt;-- level = 0
      / | \
     b  c  d        &lt;-- level = 1
   /
  f                 &lt;-- level = 2

    v1, v2, v3     
Q : b,  c,  d
    1,  1,  1

v3.d &lt;= v1.d + 1   i.e 1 &lt;= (1+1)


Now, suppose head of Q (i.e b) is removed and its neighbors are inserted in Q

    v1, v2, v3
Q : c,  d,  f
    1,  1,  2

v3.d &lt;= v1.d + 1   i.e 2 &lt;= (1+1)


This, v3.d &lt;= v1.d + 1 can be also be seen as v3.d - v1.d &lt;= 1 which means at any point of time distance between elements of queue will always be less than or equal 1. This can also be seen from the fact that during the execution of BFT only elements from 2 successive levels are present in queue. There cannot be case such that 2 elements A and B are present in queue with A at level x and B at level (x + 2).