The tree shoots in your photo could possibly be from an Elm. If you are interested in eradication and you are unable to dig out you will need to cut down and use chemical to get to the roots. Glyphosate is the best choice. In order to not damage the shrub you want to retain you must just treat the stumps of what you cut off. You can use a paint brush or rag, dipped in glyphosate, to place the chemical directly on what you want to get rid of. Be careful to not get on any plant material you want to live.