Ask Mr. Smarty Plants

From the Lady Bird Johnson Wildlife Center Native Plant Database ...

The Southern catalpa Catalpa bignonioides tree reaches 25-40 ft. in height with an equal or greater spread. Short, crooked branches form a broad, irregular crown. Its heart-shaped leaves have prominent veins and are 6–12 inches long and half as wide. The petiole is almost as long as the leaf. Deciduous leaves are large, light-green and smooth. The flowers are in clusters of 10–20, each blossom on a short stem. They are white, 2-lipped, united at the base, opening into 5 ruffled, petal-like lobes; each flower is about 2 inches across. In the throat there are 2 large yellow spots and several smaller ones, several small purple stripes, and a number of tiny purple spots. Fruit a cigarlike pod.

Some landowners and designers consider this nearly indestructible tree a nuisance because the leaves smell bad when crushed, flowers litter the ground with decaying petals after a too-brief appearance, and root suckers can create a problem. A particular green & black striped caterpillar (Sphinx moth) can completely defoliate southern catalpa trees, but the trees recover, growing new set of leaves within a month.

The USDA website lists litter and smell as the biggest management problems with ornamental catalpas. Trees drop a heavy load of flowers in the spring, then a plentiful supply of leaves in the fall, and finally a lot of large seedpods in the winter. Green leaves give off a disagreeable odor when crushed.

The tree might send up root suckers and be an issue that way, but these will be manageable and easily removed. It is unlikely that too many seedlings will be able to germinate and grow in the vicinity of the tree or in your nearby plantings unless you have a lot of bare soil in full sun (ideal seed germinating conditions). The flowers are very beautiful and look like miniature orchids on the tree and when they are newly fallen. But they will turn brown shortly after they fall and there will be some cleanup involved. The flowers do attract considerable numbers of honey bees but beyond this, it mainly provides shelter for animals.
The issues with this tree and invasiveness stem mainly from the fact that it has moved far from its native habitat in the SE US and is now showing up in California.

And lastly, the tree is weak wooded and will undoubtedly drop twigs almost continuously. When selecting your new tree look for one that has a wide branching structure - branches that are more "U" shaped instead of narrow "V"s. This will hopefully give you stronger limbs as the tree matures and less chance of breakage. Some gardeners have even trained young branches to grow horizontally to give them more strength.