diff --git a/mapstore.go b/mapstore.go
index 3db56d3..5fe381b 100644
--- a/mapstore.go
+++ b/mapstore.go
@@ -17,7 +17,7 @@ type InvalidKeyError struct {
 }
 
 func (e *InvalidKeyError) Error() string {
-	return fmt.Sprintf("invalid key: %s", e.Key)
+	return fmt.Sprintf("invalid key: %x", e.Key)
 }
 
 // SimpleMap is a simple in-memory map.
diff --git a/proofs.go b/proofs.go
index 86610a5..3a9efc4 100644
--- a/proofs.go
+++ b/proofs.go
@@ -129,11 +129,11 @@ func verifyProofWithUpdates(proof SparseMerkleProof, root []byte, key []byte, va
 	}
 
 	// Recompute root.
-	for i := len(proof.SideNodes) - 1; i >= 0; i-- {
+	for i := 0; i < len(proof.SideNodes); i++ {
 		node := make([]byte, th.pathSize())
 		copy(node, proof.SideNodes[i])
 
-		if hasBit(path, i) == right {
+		if getBitAtFromMSB(path, len(proof.SideNodes)-1-i) == right {
 			currentHash, currentData = th.digestNode(node, currentHash)
 		} else {
 			currentHash, currentData = th.digestNode(currentHash, node)
@@ -170,7 +170,7 @@ func CompactProof(proof SparseMerkleProof, hasher hash.Hash) (SparseCompactMerkl
 		node := make([]byte, th.hasher.Size())
 		copy(node, proof.SideNodes[i])
 		if bytes.Equal(node, th.placeholder()) {
-			setBit(bitMask, i)
+			setBitAtFromMSB(bitMask, i)
 		} else {
 			compactedSideNodes = append(compactedSideNodes, node)
 		}
@@ -195,7 +195,7 @@ func DecompactProof(proof SparseCompactMerkleProof, hasher hash.Hash) (SparseMer
 	decompactedSideNodes := make([][]byte, proof.NumSideNodes)
 	position := 0
 	for i := 0; i < proof.NumSideNodes; i++ {
-		if hasBit(proof.BitMask, i) == 1 {
+		if getBitAtFromMSB(proof.BitMask, i) == 1 {
 			decompactedSideNodes[i] = th.placeholder()
 		} else {
 			decompactedSideNodes[i] = proof.SideNodes[position]
diff --git a/proofs_test.go b/proofs_test.go
index f0be57b..893262e 100644
--- a/proofs_test.go
+++ b/proofs_test.go
@@ -210,7 +210,7 @@ func TestCompactProofsSanityCheck(t *testing.T) {
 
 	// Case (compact proofs): unexpected bit mask length.
 	proof, _ = smt.ProveCompact([]byte("testKey1"))
-	proof.NumSideNodes = 1
+	proof.NumSideNodes = 10
 	if proof.sanityCheck(th) {
 		t.Error("sanity check incorrectly passed")
 	}
@@ -221,7 +221,7 @@ func TestCompactProofsSanityCheck(t *testing.T) {
 
 	// Case (compact proofs): unexpected number of sidenodes for number of side nodes.
 	proof, _ = smt.ProveCompact([]byte("testKey1"))
-	proof.SideNodes = proof.SideNodes[:1]
+	proof.SideNodes = append(proof.SideNodes, proof.SideNodes...)
 	if proof.sanityCheck(th) {
 		t.Error("sanity check incorrectly passed")
 	}
diff --git a/smt.go b/smt.go
index 2882426..99edfec 100644
--- a/smt.go
+++ b/smt.go
@@ -96,7 +96,7 @@ func (smt *SparseMerkleTree) GetForRoot(key []byte, root []byte) ([]byte, error)
 		}
 
 		leftNode, rightNode := smt.th.parseNode(currentData)
-		if hasBit(path, i) == right {
+		if getBitAtFromMSB(path, i) == right {
 			currentHash = rightNode
 		} else {
 			currentHash = leftNode
@@ -188,7 +188,7 @@ func (smt *SparseMerkleTree) deleteWithSideNodes(path []byte, sideNodes [][]byte
 
 	var currentHash, currentData []byte
 	nonPlaceholderReached := false
-	for i := smt.depth() - 1; i >= 0; i-- {
+	for i := 0; i < len(sideNodes); i++ {
 		if sideNodes[i] == nil {
 			continue
 		}
@@ -215,7 +215,7 @@ func (smt *SparseMerkleTree) deleteWithSideNodes(path []byte, sideNodes [][]byte
 		}
 
 		if !nonPlaceholderReached && bytes.Equal(sideNode, smt.th.placeholder()) {
-			// We found another placeholder sibling node, keep going down the
+			// We found another placeholder sibling node, keep going up the
 			// tree until we find the first sibling that is not a placeholder.
 			continue
 		} else if !nonPlaceholderReached {
@@ -224,7 +224,7 @@ func (smt *SparseMerkleTree) deleteWithSideNodes(path []byte, sideNodes [][]byte
 			nonPlaceholderReached = true
 		}
 
-		if hasBit(path, i) == right {
+		if getBitAtFromMSB(path, len(sideNodes)-1-i) == right {
 			currentHash, currentData = smt.th.digestNode(sideNode, currentData)
 		} else {
 			currentHash, currentData = smt.th.digestNode(currentData, sideNode)
@@ -269,7 +269,7 @@ func (smt *SparseMerkleTree) updateWithSideNodes(path []byte, value []byte, side
 		commonPrefixCount = countCommonPrefix(path, actualPath)
 	}
 	if commonPrefixCount != smt.depth() {
-		if hasBit(path, commonPrefixCount) == right {
+		if getBitAtFromMSB(path, commonPrefixCount) == right {
 			currentHash, currentData = smt.th.digestNode(oldLeafHash, currentData)
 		} else {
 			currentHash, currentData = smt.th.digestNode(currentData, oldLeafHash)
@@ -283,11 +283,15 @@ func (smt *SparseMerkleTree) updateWithSideNodes(path []byte, value []byte, side
 		currentData = currentHash
 	}
 
-	for i := smt.depth() - 1; i >= 0; i-- {
+	for i := 0; i < smt.depth(); i++ {
 		sideNode := make([]byte, smt.th.pathSize())
 
-		if sideNodes[i] == nil {
-			if commonPrefixCount != smt.depth() && commonPrefixCount > i {
+		// The offset from the bottom of the tree to the start of the side nodes
+		// i-offsetOfSideNodes is the index into sideNodes[]
+		offsetOfSideNodes := smt.depth() - len(sideNodes)
+
+		if i-offsetOfSideNodes < 0 || sideNodes[i-offsetOfSideNodes] == nil {
+			if commonPrefixCount != smt.depth() && commonPrefixCount > smt.depth()-1-i {
 				// If there are no sidenodes at this height, but the number of
 				// bits that the paths of the two leaf nodes share in common is
 				// greater than this height, then we need to build up the tree
@@ -297,10 +301,10 @@ func (smt *SparseMerkleTree) updateWithSideNodes(path []byte, value []byte, side
 				continue
 			}
 		} else {
-			copy(sideNode, sideNodes[i])
+			copy(sideNode, sideNodes[i-offsetOfSideNodes])
 		}
 
-		if hasBit(path, i) == right {
+		if getBitAtFromMSB(path, smt.depth()-1-i) == right {
 			currentHash, currentData = smt.th.digestNode(sideNode, currentData)
 		} else {
 			currentHash, currentData = smt.th.digestNode(currentData, sideNode)
@@ -319,7 +323,9 @@ func (smt *SparseMerkleTree) updateWithSideNodes(path []byte, value []byte, side
 // Returns an array of sibling nodes, the leaf hash found at that path and the
 // leaf data. If the leaf is a placeholder, the leaf data is nil.
 func (smt *SparseMerkleTree) sideNodesForRoot(path []byte, root []byte) ([][]byte, []byte, []byte, error) {
-	sideNodes := make([][]byte, smt.depth())
+	// Side nodes for the path. Nodes are inserted in reverse order, then the
+	// slice is reversed at the end.
+	sideNodes := make([][]byte, 0, smt.depth())
 
 	if bytes.Equal(root, smt.th.placeholder()) {
 		// If the root is a placeholder, there are no sidenodes to return.
@@ -340,17 +346,17 @@ func (smt *SparseMerkleTree) sideNodesForRoot(path []byte, root []byte) ([][]byt
 		leftNode, rightNode := smt.th.parseNode(currentData)
 
 		// Get sidenode depending on whether the path bit is on or off.
-		if hasBit(path, i) == right {
-			sideNodes[i] = leftNode
+		if getBitAtFromMSB(path, i) == right {
+			sideNodes = append(sideNodes, leftNode)
 			nodeHash = rightNode
 		} else {
-			sideNodes[i] = rightNode
+			sideNodes = append(sideNodes, rightNode)
 			nodeHash = leftNode
 		}
 
 		if bytes.Equal(nodeHash, smt.th.placeholder()) {
 			// If the node is a placeholder, we've reached the end.
-			return sideNodes, nodeHash, nil, nil
+			return reverseSideNodes(sideNodes), nodeHash, nil, nil
 		}
 
 		currentData, err = smt.ms.Get(nodeHash)
@@ -362,7 +368,7 @@ func (smt *SparseMerkleTree) sideNodesForRoot(path []byte, root []byte) ([][]byt
 		}
 	}
 
-	return sideNodes, nodeHash, currentData, err
+	return reverseSideNodes(sideNodes), nodeHash, currentData, nil
 }
 
 // Prove generates a Merkle proof for a key.
diff --git a/smt_test.go b/smt_test.go
index 1b9509a..41bfedc 100644
--- a/smt_test.go
+++ b/smt_test.go
@@ -168,7 +168,117 @@ func TestSparseMerkleTreeUpdateBasic(t *testing.T) {
 	}
 }
 
-// Test tree operations when two leafs are immediate neighbours.
+// Test known tree ops
+func TestSparseMerkleTreeKnown(t *testing.T) {
+	h := newDummyHasher(sha256.New())
+	sm := NewSimpleMap()
+	smt := NewSparseMerkleTree(sm, h)
+	var value []byte
+	var err error
+
+	baseKey := make([]byte, h.Size()+4)
+	key1 := make([]byte, h.Size()+4)
+	copy(key1, baseKey)
+	key1[4] = byte(0b00000000)
+	key2 := make([]byte, h.Size()+4)
+	copy(key2, baseKey)
+	key2[4] = byte(0b01000000)
+	key3 := make([]byte, h.Size()+4)
+	copy(key3, baseKey)
+	key3[4] = byte(0b10000000)
+	key4 := make([]byte, h.Size()+4)
+	copy(key4, baseKey)
+	key4[4] = byte(0b11000000)
+	key5 := make([]byte, h.Size()+4)
+	copy(key5, baseKey)
+	key5[4] = byte(0b11010000)
+
+	_, err = smt.Update(key1, []byte("testValue1"))
+	if err != nil {
+		t.Errorf("returned error when updating empty key: %v", err)
+	}
+	_, err = smt.Update(key2, []byte("testValue2"))
+	if err != nil {
+		t.Errorf("returned error when updating empty key: %v", err)
+	}
+	_, err = smt.Update(key3, []byte("testValue3"))
+	if err != nil {
+		t.Errorf("returned error when updating empty key: %v", err)
+	}
+	_, err = smt.Update(key4, []byte("testValue4"))
+	if err != nil {
+		t.Errorf("returned error when updating empty key: %v", err)
+	}
+	_, err = smt.Update(key5, []byte("testValue5"))
+	if err != nil {
+		t.Errorf("returned error when updating empty key: %v", err)
+	}
+
+	value, err = smt.Get(key1)
+	if err != nil {
+		t.Errorf("returned error when getting non-empty key: %v", err)
+	}
+	if !bytes.Equal([]byte("testValue1"), value) {
+		t.Error("did not get correct value when getting non-empty key")
+	}
+	value, err = smt.Get(key2)
+	if err != nil {
+		t.Errorf("returned error when getting non-empty key: %v", err)
+	}
+	if !bytes.Equal([]byte("testValue2"), value) {
+		t.Error("did not get correct value when getting non-empty key")
+	}
+	value, err = smt.Get(key3)
+	if err != nil {
+		t.Errorf("returned error when getting non-empty key: %v", err)
+	}
+	if !bytes.Equal([]byte("testValue3"), value) {
+		t.Error("did not get correct value when getting non-empty key")
+	}
+	value, err = smt.Get(key4)
+	if err != nil {
+		t.Errorf("returned error when getting non-empty key: %v", err)
+	}
+	if !bytes.Equal([]byte("testValue4"), value) {
+		t.Error("did not get correct value when getting non-empty key")
+	}
+	value, err = smt.Get(key5)
+	if err != nil {
+		t.Errorf("returned error when getting non-empty key: %v", err)
+	}
+	if !bytes.Equal([]byte("testValue5"), value) {
+		t.Error("did not get correct value when getting non-empty key")
+	}
+
+	proof1, _ := smt.Prove(key1)
+	proof2, _ := smt.Prove(key2)
+	proof3, _ := smt.Prove(key3)
+	proof4, _ := smt.Prove(key4)
+	proof5, _ := smt.Prove(key5)
+	dsmst := NewDeepSparseMerkleSubTree(NewSimpleMap(), h, smt.Root())
+	err = dsmst.AddBranch(proof1, key1, []byte("testValue1"))
+	if err != nil {
+		t.Errorf("returned error when adding branch to deep subtree: %v", err)
+	}
+	err = dsmst.AddBranch(proof2, key2, []byte("testValue2"))
+	if err != nil {
+		t.Errorf("returned error when adding branch to deep subtree: %v", err)
+	}
+	err = dsmst.AddBranch(proof3, key3, []byte("testValue3"))
+	if err != nil {
+		t.Errorf("returned error when adding branch to deep subtree: %v", err)
+	}
+	err = dsmst.AddBranch(proof4, key4, []byte("testValue4"))
+	if err != nil {
+		t.Errorf("returned error when adding branch to deep subtree: %v", err)
+	}
+	err = dsmst.AddBranch(proof5, key5, []byte("testValue5"))
+	if err != nil {
+		t.Errorf("returned error when adding branch to deep subtree: %v", err)
+	}
+}
+
+// Test tree operations when two leafs are immediate neighbors.
 func TestSparseMerkleTreeMaxHeightCase(t *testing.T) {
 	h := newDummyHasher(sha256.New())
 	sm := NewSimpleMap()
@@ -176,9 +286,9 @@ func TestSparseMerkleTreeMaxHeightCase(t *testing.T) {
 	var value []byte
 	var err error
 
-	// Make two neighbouring keys.
+	// Make two neighboring keys.
 	//
-	// The dummy hash function excepts keys to prefixed with four bytes of 0,
+	// The dummy hash function expects keys to prefixed with four bytes of 0,
 	// which will cause it to return the preimage itself as the digest, without
 	// the first four bytes.
 	key1 := make([]byte, h.Size()+4)
@@ -187,7 +297,8 @@ func TestSparseMerkleTreeMaxHeightCase(t *testing.T) {
 	key1[h.Size()+4-1] = byte(0)
 	key2 := make([]byte, h.Size()+4)
 	copy(key2, key1)
-	setBit(key2, (h.Size()+4)*8-1)
+	// We make key2's least significant bit different than key1's
+	key2[h.Size()+4-1] = byte(1)
 
 	_, err = smt.Update(key1, []byte("testValue1"))
 	if err != nil {
@@ -205,7 +316,6 @@ func TestSparseMerkleTreeMaxHeightCase(t *testing.T) {
 	if !bytes.Equal([]byte("testValue1"), value) {
 		t.Error("did not get correct value when getting non-empty key")
 	}
-
 	value, err = smt.Get(key2)
 	if err != nil {
 		t.Errorf("returned error when getting non-empty key: %v", err)
@@ -213,6 +323,14 @@ func TestSparseMerkleTreeMaxHeightCase(t *testing.T) {
 	if !bytes.Equal([]byte("testValue2"), value) {
 		t.Error("did not get correct value when getting non-empty key")
 	}
+
+	proof, err := smt.Prove(key1)
+	if err != nil {
+		t.Errorf("returned error when proving key: %v", err)
+	}
+	if len(proof.SideNodes) != 256 {
+		t.Errorf("unexpected proof size")
+	}
 }
 
 // Test base case tree delete operations with a few keys.
diff --git a/utils.go b/utils.go
index d413bfd..55163cd 100644
--- a/utils.go
+++ b/utils.go
@@ -1,22 +1,24 @@
 package smt
 
-func hasBit(data []byte, position int) int {
-	if int(data[position/8])&(1<<(uint(position)%8)) > 0 {
+// getBitAtFromMSB gets the bit at an offset from the most significant bit
+func getBitAtFromMSB(data []byte, position int) int {
+	if int(data[position/8])&(1<<(8-1-uint(position)%8)) > 0 {
 		return 1
 	}
 	return 0
 }
 
-func setBit(data []byte, position int) {
+// setBitAtFromMSB sets the bit at an offset from the most significant bit
+func setBitAtFromMSB(data []byte, position int) {
 	n := int(data[position/8])
-	n |= (1 << (uint(position) % 8))
+	n |= (1 << (8 - 1 - uint(position)%8))
 	data[position/8] = byte(n)
 }
 
 func countSetBits(data []byte) int {
 	count := 0
 	for i := 0; i < len(data)*8; i++ {
-		if hasBit(data, i) == 1 {
+		if getBitAtFromMSB(data, i) == 1 {
 			count++
 		}
 	}
@@ -26,7 +28,7 @@ func countSetBits(data []byte) int {
 func countCommonPrefix(data1 []byte, data2 []byte) int {
 	count := 0
 	for i := 0; i < len(data1)*8; i++ {
-		if hasBit(data1, i) == hasBit(data2, i) {
+		if getBitAtFromMSB(data1, i) == getBitAtFromMSB(data2, i) {
 			count++
 		} else {
 			break
@@ -39,3 +41,11 @@ func emptyBytes(length int) []byte {
 	b := make([]byte, length)
 	return b
 }
+
+func reverseSideNodes(sideNodes [][]byte) [][]byte {
+	for left, right := 0, len(sideNodes)-1; left < right; left, right = left+1, right-1 {
+		sideNodes[left], sideNodes[right] = sideNodes[right], sideNodes[left]
+	}
+
+	return sideNodes
+}