It is very common that grafted trees produce suckers- this being the commonly used name for vigorous shoots coming from the rootstock of the plant. In some trees such as plums these are often produced for many years after planting and have to be removed regularly.
With Cercis it is also quite likely that your tree will put out suckers from the base of the plant below the graft especially in the first few years. In this event they should be gently removed, best by bending the shoots back so it snaps away from the main stem with only a small wound. Pruning the shoot will just allow it to re-grow and then it will be even more vigorous.
There are three main issues with suckers. Firstly they are not of the cultivar you require and so look and perform very differently. Secondly, they take energy away from the main growth and thus the tree may lack vigor. Thirdly, and often overlooked, if herbicide is used to keep the base of the tree clean from weeds then the chemical can get onto the suckers and this can damage or even kill the tree.
The best time to take off the suckers is in early spring as the tree is just starting to break dormancy. In the unfortunate event of your tree dying then it is likely the rootstock will put out many suckers. Gardeners can be led to believe that this is their tree coming back to life, but in reality it is an illusion- the tree is finished and should be replaced.