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This sounds like a fungal infection. Contact your local extension office (found under gov't in the phone book) for more information on how to control this problem. You may also find this article helpful:

https://www.gardeningknowhow.com/ornamental/trees/eucalyptus/eucalyptus-tree-problems.htm

Your trees may very well be diseased. Please see these articles for more information:

https://www.gardeningknowhow.com/ornamental/trees/eucalyptus/eucalyptus-tree-problems.htm

https://www.gardeningknowhow.com/ornamental/trees/eucalyptus/tips-growing-eucalyptus.htm

You might want to try the technique of pollarding the silver dollar:
http://blog.metmuseum.org/cloistersgardens/2011/02/25/woodswoman-pollard-that-tree/

And more about Eucalyptus here:
https://www.gardeningknowhow.com/ornamental/trees/eucalyptus/tips-growing-eucalyptus.htm

Generally, indoor eucalyptus rarely flower. To enjoy flowers, you'll have to grow the tree outdoors in the landscape.

Applying wound dressing is only needed for larger areas to prevent infection and even this isn't always necessary as trees often 'heal' themselves. A light coating of a wound dressing might also be used for cosmetic purposes. However, you may use wound dressings if you wish. Look for a breathable wound dressing for pruned, grafted or damaged plants found in garden, landscaping, or hardware stores. For better healing results, cut the branches flush with the tree.

The tree may have a pest problem and the pests are secreting honeydew. It would be almost the consistency of water. Honeydew, though, attracts fungus, so what you may be seeing is a fungus growing on the deck because of honeydew. Check the tree carefully for pests. They will most likely be under leaves or in the crotch area of branches.