Tree codes

For the codes generated in the tree isomorphism testing algorithm, decide if it can exist:

Variant
A rooted tree code that contains the sequence 0000 but does not contain 111.
Hint
First, find the code and see if there is a tree for it.
Answer
Yes, e.g. 000011011011.
Variant
A code of a cultivated tree, which contains the sequence 1111 but does not contain 000.
Answer
Yes, e.g. 001001001111.
Variant
A rooted tree code, which contains the sequence 1111 but does not contain 000.
Solution
Due to the lexicographic sorting, the initial sequence of zeros corresponds to the longest path from the root, while each section of ones corresponds to some path towards the root.
Answer
No, there is no such code.