The complete binary tree is an important network structure for parallel and distributed computing, which has many nice properties and is often used to be embedded into other interconnection architectures. The locally twisted cube LTQ(n) is an important variant of the hypercube Q(n). It has many better properties than Q(n) with the same number of edges and vertices. The main results obtained in this paper are: (1) The complete binary tree CBTn rooted at an arbitrary vertex of LTQ(n) can be embedded with dilation 2 and congestion I into LTQ(n). (2) When there exists only one faulty node in LTQ(n) both the dilation and congestion will become 2 after reconfiguring CBTn. (3) When there exist two faulty nodes in LTQ(n) then both the dilation and congestion will become 3 after reconfiguring CBTn. (4) For any faulty set F of LTQ(n) with 2 < vertical bar F vertical bar <= 2(n-1), both the dilation and congestion will become 3 after reconfiguring CBTn, under certain constraints. (C) 2016 Elsevier Inc. All rights reserved.